RIMS
Kôkyûroku
Bessatsu
Bx
(2020),
000–000
The
Mathematics
of
Mutually
Alien
Copies:
from
Gaussian
Integrals
to
Inter-universal
Teichmüller
Theory
By
Shinichi
Mochizuki
Abstract
Inter-universal
Teichmüller
theory
may
be
described
as
a
construction
of
certain
canonical
deformations
of
the
ring
structure
of
a
number
field
equipped
with
certain
auxiliary
data,
which
includes
an
elliptic
curve
over
the
number
field
and
a
prime
number
≥
5.
In
the
present
paper,
we
survey
this
theory
by
focusing
on
the
rich
analogies
between
this
theory
and
the
classical
computation
of
the
Gaussian
integral.
The
main
common
features
that
underlie
these
analogies
may
be
summarized
as
follows:
·
the
introduction
of
two
mutually
alien
copies
of
the
object
of
interest;
·
the
computation
of
the
effect
—
i.e.,
on
the
two
mutually
alien
copies
of
the
object
of
interest
—
of
two-dimensional
changes
of
coordinates
by
considering
the
effect
on
infinitesimals;
·
the
passage
from
planar
cartesian
to
polar
coordinates
and
the
resulting
split-
ting,
or
decoupling,
into
radial
—
i.e.,
in
more
abstract
valuation-theoretic
termi-
nology,
“value
group”
—
and
angular
—
i.e.,
in
more
abstract
valuation-theoretic
terminology,
“unit
group”
—
portions;
·
the
straightforward
evaluation
of
the
radial
portion
by
applying
the
quadraticity
of
the
exponent
of
the
Gaussian
distribution;
·
the
straightforward
evaluation
of
the
angular
portion
by
considering
the
met-
ric
geometry
of
the
group
of
units
determined
by
a
suitable
version
of
the
natural
logarithm
function.
[Here,
the
intended
sense
of
the
descriptive
“alien”
is
that
of
its
original
Latin
root,
i.e.,
a
sense
of
abstract,
tautological
“otherness”.]
After
reviewing
the
classical
computation
of
the
Gaussian
integral,
we
give
a
detailed
survey
of
inter-universal
Teichmüller
theory
by
concentrating
on
the
common
features
listed
above.
The
paper
concludes
with
a
discussion
of
various
historical
aspects
of
the
mathematics
that
appears
in
inter-universal
Teichmüller
theory.
Received
xxxx
xx,
2016.
Revised
xxxx
xx,
2020.
2010
Mathematics
Subject
Classification(s):
Primary:
14H25;
Secondary:
14H30.
c
2020
Research
Institute
for
Mathematical
Sciences,
Kyoto
University.
All
rights
reserved.
2
Shinichi
Mochizuki
Contents
§
1.
Review
of
the
computation
of
the
Gaussian
integral
§
1.1.
Inter-universal
Teichmüller
theory
via
the
Gaussian
integral
§
1.2.
Naive
approach
via
changes
of
coordinates
or
partial
integrations
§
1.3.
Introduction
of
identical
but
mutually
alien
copies
§
1.4.
Integrals
over
two-dimensional
Euclidean
space
§
1.5.
The
effect
on
infinitesimals
of
changes
of
coordinates
§
1.6.
Passage
from
planar
cartesian
to
polar
coordinates
§
1.7.
Justification
of
naive
approach
up
to
an
“error
factor”
§
2.
Changes
of
universe
as
arithmetic
changes
of
coordinates
§
2.1.
The
issue
of
bounding
heights:
the
ABC
and
Szpiro
Conjectures
§
2.2.
Arithmetic
degrees
as
global
integrals
§
2.3.
Bounding
heights
via
global
multiplicative
subspaces
§
2.4.
Bounding
heights
via
Frobenius
morphisms
on
number
fields
§
2.5.
Fundamental
example
of
the
derivative
of
a
Frobenius
lifting
§
2.6.
Positive
characteristic
model
for
mono-anabelian
transport
§
2.7.
The
apparatus
and
terminology
of
mono-anabelian
transport
§
2.8.
Remark
on
the
usage
of
certain
terminology
§
2.9.
Mono-anabelian
transport
and
the
Kodaira-Spencer
morphism
§
2.10.
Inter-universality:
changes
of
universe
as
changes
of
coordinates
§
2.11.
The
two
underlying
combinatorial
dimensions
of
a
ring
§
2.12.
Mono-anabelian
transport
for
mixed-characteristic
local
fields
§
2.13.
Mono-anabelian
transport
for
monoids
of
rational
functions
§
2.14.
Finite
discrete
approximations
of
harmonic
analysis
§
3.
Multiradiality:
an
abstract
analogue
of
parallel
transport
§
3.1.
The
notion
of
multiradiality
§
3.2.
Fundamental
examples
of
multiradiality
§
3.3.
The
log-theta-lattice:
Θ
±ell
N
F
-Hodge
theaters,
log-links,
Θ-links
§
3.4.
Kummer
theory
and
multiradial
decouplings/cyclotomic
rigidity
§
3.5.
Remarks
on
the
use
of
Frobenioids
§
3.6.
Galois
evaluation,
labels,
symmetries,
and
log-shells
§
3.7.
Log-volume
estimates
via
the
multiradial
representation
§
3.8.
Comparison
with
the
Gaussian
integral
§
3.9.
Relation
to
scheme-theoretic
Hodge-Arakelov
theory
§
3.10.
The
technique
of
tripodal
transport
§
3.11.
Mathematical
analysis
of
elementary
conceptual
discomfort
§
4.
Historical
comparisons
and
analogies
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
§
4.1.
§
4.2.
§
4.3.
§
4.4.
3
Numerous
connections
to
classical
theories
Contrasting
aspects
of
class
field
theory
and
Kummer
theory
Arithmetic
and
geometric
versions
of
the
Mordell
Conjecture
Atavistic
resemblance
in
the
development
of
mathematics
Introduction
In
the
present
paper,
we
survey
inter-universal
Teichmüller
theory
by
focusing
on
the
rich
analogies
[cf.
§3.8]
between
this
theory
and
the
classical
computation
of
the
Gaussian
integral.
Inter-universal
Teichmüller
theory
concerns
the
construction
of
canonical
deformations
of
the
ring
structure
of
a
number
field
equipped
with
certain
auxiliary
data.
The
collection
of
data,
i.e.,
consisting
of
the
number
field
equipped
with
certain
auxiliary
data,
to
which
inter-universal
Teichmüller
theory
is
applied
is
referred
to
as
initial
Θ-data
[cf.
§3.3,
(i),
for
more
details].
The
principal
components
of
a
collection
of
initial
Θ-data
are
·
the
given
number
field,
·
an
elliptic
curve
over
the
number
field,
and
·
a
prime
number
l
≥
5.
The
main
applications
of
inter-universal
Teichmüller
theory
to
diophantine
geom-
etry
[cf.
§3.7,
(iv),
for
more
details]
are
obtained
by
applying
the
canonical
deformation
constructed
for
a
specific
collection
of
initial
Θ-data
to
bound
the
height
of
the
elliptic
curve
that
appears
in
the
initial
Θ-data.
Let
N
be
a
fixed
natural
number
>
1.
Then
the
issue
of
bounding
a
given
non-
negative
real
number
h
∈
R
≥0
may
be
understood
as
the
issue
of
showing
that
N
·
h
is
roughly
equal
to
h,
i.e.,
N
·
h
“≈”
h
[cf.
§2.3,
§2.4].
When
h
is
the
height
of
an
elliptic
curve
over
a
number
field,
this
issue
may
be
understood
as
the
issue
of
showing
that
the
height
of
the
[in
fact,
in
most
cases,
fictional!]
“elliptic
curve”
whose
q-parameters
are
the
N
-th
powers
“q
N
”
of
the
q-parameters
“q”
of
the
given
elliptic
curve
is
roughly
equal
to
the
height
of
the
given
elliptic
curve,
i.e.,
that,
at
least
from
the
point
of
view
of
[global]
heights,
q
N
“≈”
q
[cf.
§2.3,
§2.4].
In
order
to
verify
the
approximate
relation
q
N
“≈”
q,
one
begins
by
introducing
two
distinct
—
i.e.,
two
“mutually
alien”
—
copies
of
the
conventional
scheme
Shinichi
Mochizuki
4
theory
surrounding
the
given
initial
Θ-data.
Here,
the
intended
sense
of
the
descriptive
“alien”
is
that
of
its
original
Latin
root,
i.e.,
a
sense
of
abstract,
tautological
“otherness”.
These
two
mutually
alien
copies
of
conventional
scheme
theory
are
glued
together
—
by
considering
relatively
weak
underlying
structures
of
the
respective
conventional
scheme
theories
such
as
multiplicative
monoids
and
profinite
groups
—
in
such
a
way
that
the
“q
N
”
in
one
copy
of
scheme
theory
is
identified
with
the
“q”
in
the
other
copy
of
scheme
theory.
This
gluing
is
referred
to
as
the
Θ-link.
Thus,
the
“q
N
”
on
the
left-hand
side
of
the
Θ-link
is
glued
to
the
“q”
on
the
right-hand
side
of
the
Θ-link,
i.e.,
N
q
LHS
“=”
q
RHS
[cf.
§3.3,
(vii),
for
more
details].
Here,
“N
”
is
in
fact
taken
not
to
be
a
fixed
natural
number,
but
rather
a
sort
of
symmetrized
average
over
the
values
j
2
,
where
j
=
def
1,
.
.
.
,
l
,
and
we
write
l
=
(l
−
1)/2.
Thus,
the
left-hand
side
of
the
above
display
2
j
{q
LHS
}
j
bears
a
striking
formal
resemblance
to
the
Gaussian
distribution.
One
then
verifies
the
desired
approximate
relation
q
N
“≈”
q
by
computing
2
j
{q
LHS
}
j
—
not
in
terms
of
q
LHS
[which
is
immediate
from
the
definitions!],
but
rather
—
in
terms
of
[the
scheme
theory
surrounding]
q
RHS
[which
is
a
highly
nontrivial
matter!].
The
conclusion
of
this
computation
may
be
sum-
marized
as
follows:
up
to
relatively
mild
indeterminacies
—
i.e.,
“relatively
small
error
terms”
j
2
j
2
}
j
may
be
“confused”,
or
“identified”,
with
{q
RHS
}
j
,
that
is
to
say,
—
{q
LHS
2
j
}
j
{q
LHS
(“=”
q
RHS
)
!!
2
j
{q
RHS
}
j
[cf.
the
discussion
of
§3.7,
(i)
especially,
Fig.
3.19,
as
well
as
the
discussion
of
§3.10,
(ii),
and
§3.11,
(iv),
(v),
for
more
details].
Once
one
is
equipped
with
this
“license”
j
2
j
2
}
j
with
{q
RHS
}
j
,
the
derivation
of
the
desired
approximate
to
confuse/identify
{q
LHS
relation
2
{q
j
}
j
“≈”
q
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
5
and
hence
of
the
desired
bounds
on
heights
is
an
essentially
formal
matter
[cf.
§3.7,
(ii),
(iv);
§3.11,
(iv),
(v)].
The
starting
point
of
the
exposition
of
the
present
paper
lies
in
the
observation
[cf.
§3.8
for
more
details]
that
the
main
features
of
the
theory
underlying
the
computation
j
2
}
j
in
terms
of
q
RHS
exhibit
remarkable
similarities
—
as
is
just
discussed
of
{q
LHS
perhaps
foreshadowed
by
the
striking
formal
resemblance
observed
above
to
the
Gaus-
sian
distribution
—
to
the
main
features
of
the
classical
computation
of
the
Gaussian
integral,
namely,
(1
mf
)
the
introduction
of
two
mutually
alien
copies
of
the
object
of
interest
[cf.
§3.8,
gau
gau
(1
),
(2
)];
(2
mf
)
the
computation
of
the
effect
—
i.e.,
on
the
two
mutually
alien
copies
of
the
object
of
interest
—
of
two-dimensional
changes
of
coordinates
by
considering
gau
gau
gau
gau
the
effect
on
infinitesimals
[cf.
§3.8,
(3
),
(4
),
(5
),
(6
)];
(3
mf
)
the
passage
from
planar
cartesian
to
polar
coordinates
and
the
resulting
splitting,
or
decoupling,
into
radial
—
i.e.,
in
more
abstract
valuation-theoretic
terminology,
“value
group”
—
and
angular
—
i.e.,
in
more
abstract
valuation-
gau
gau
theoretic
terminology,
“unit
group”
—
portions
[cf.
§3.8,
(7
),
(8
)];
(4
mf
)
the
straightforward
evaluation
of
the
radial
portion
by
applying
the
quadratic-
gau
gau
ity
of
the
exponent
of
the
Gaussian
distribution
[cf.
§3.8,
(9
),
(11
)];
(5
mf
)
the
straightforward
evaluation
of
the
angular
portion
by
considering
the
metric
geometry
of
the
group
of
units
determined
by
a
suitable
version
of
the
natural
gau
gau
logarithm
function
[cf.
§3.8,
(10
),
(11
)].
In
passing,
we
mention
that
yet
another
brief
overview
of
certain
important
aspects
of
inter-universal
Teichmüller
theory
from
a
very
elementary
point
of
view
may
be
found
in
§3.11.
The
present
paper
begins,
in
§1,
with
a
review
of
the
classical
computation
of
the
Gaussian
integral,
by
breaking
down
this
familiar
computation
into
steps
in
such
a
way
as
to
facilitate
the
subsequent
comparison
with
inter-universal
Teichmüller
theory.
We
then
proceed,
in
§2,
to
discuss
the
portion
of
inter-universal
Teichmüller
theory
mf
that
corresponds
to
(2
).
The
exposition
of
§2
was
designed
so
as
to
be
accessible
to
readers
familiar
with
well-known
portions
of
scheme
theory
and
the
theory
of
the
étale
fundamental
group
—
i.e.,
at
the
level
of
[Harts]
and
[SGA1].
The
various
Examples
that
appear
in
this
exposition
of
§2
include
numerous
well-defined
and
relatively
straightforward
mathematical
assertions
Shinichi
Mochizuki
6
often
without
complete
proofs.
In
particular,
the
reader
may
think
of
the
task
of
supplying
a
complete
proof
for
any
of
these
assertions
as
a
sort
of
“exercise”
and
hence
of
§2
itself
as
a
sort
of
workbook
with
exercises.
At
the
level
of
papers,
§2
is
concerned
mainly
with
the
content
of
the
“classical”
pa-
per
[Uchi]
of
Uchida
and
the
“preparatory
papers”
[FrdI],
[FrdII],
[GenEll],
[AbsTopI],
[AbsTopII],
[AbsTopIII].
By
contrast,
the
level
of
exposition
of
§3
is
substantially
less
elementary
than
that
of
§2.
In
§3,
we
apply
the
conceptual
infrastructure
exposed
in
§2
to
survey
those
aspects
of
inter-universal
Teichmüller
theory
that
correspond
to
mf
mf
mf
mf
(1
),
(3
),
(4
),
and
(5
),
i.e.,
at
the
level
of
papers,
to
[EtTh],
[IUTchI],
[IUTchII],
[IUTchIII],
[IUTchIV].
Finally,
in
§4,
we
reflect
on
various
historical
aspects
of
the
theory
exposed
in
§2
and
§3.
Acknowledgements:
The
author
wishes
to
express
his
appreciation
for
the
stimulating
comments
that
he
has
received
from
numerous
mathematicians
concerning
the
theory
exposed
in
the
present
paper
and,
especially,
his
deep
gratitude
to
Fumiharu
Kato,
Akio
Tamagawa,
Go
Yamashita,
Mohamed
Saı̈di,
Yuichiro
Hoshi,
Ivan
Fesenko,
Fucheng
Tan,
Emmanuel
Lepage,
Arata
Minamide,
and
Wojciech
Porowski
for
the
very
active
and
devoted
role
that
they
played
both
in
discussing
this
theory
with
the
author
and
in
disseminating
it
to
others.
In
particular,
the
author
would
like
to
thank
Yuichiro
Hoshi
for
introducing
the
notion
of
mono-anabelian
transport
as
a
means
of
formulating
a
technique
that
is
frequently
applied
throughout
the
theory.
This
notion
plays
a
central
role
in
the
expository
approach
adopted
in
the
present
paper.
§
1.
§
1.1.
Review
of
the
computation
of
the
Gaussian
integral
Inter-universal
Teichmüller
theory
via
the
Gaussian
integral
The
goal
of
the
present
paper
is
to
pave
the
road,
for
the
reader,
from
a
state
of
complete
ignorance
of
inter-universal
Teichmüller
theory
to
a
state
of
general
appreci-
ation
of
the
“game
plan”
of
inter-universal
Teichmüller
theory
by
reconsidering
the
well-known
computation
of
the
Gaussian
integral
∞
√
2
e
−x
dx
=
π
−∞
via
polar
coordinates
from
the
point
of
view
of
a
hypothetical
high-school
student
who
has
studied
one-variable
calculus
and
polar
coordinates,
but
has
not
yet
had
any
ex-
posure
to
multi-variable
calculus.
That
is
to
say,
we
shall
begin
in
the
present
§1
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
7
by
reviewing
this
computation
of
the
Gaussian
integral
by
discussing
how
this
compu-
tation
might
be
explained
to
such
a
hypothetical
high-school
student.
In
subsequent
§’s,
we
then
proceed
to
discuss
how
various
key
steps
in
such
an
explanation
to
a
hypothetical
high-school
student
may
be
translated
into
the
more
sophisticated
lan-
guage
of
abstract
arithmetic
geometry
in
such
a
way
as
to
yield
a
general
outline
of
inter-universal
Teichmüller
theory
based
on
the
deep
structural
similarities
between
inter-universal
Teichmüller
theory
and
the
computation
of
the
Gaussian
integral.
§
1.2.
Naive
approach
via
changes
of
coordinates
or
partial
integrations
In
one-variable
calculus,
definite
integrals
that
appear
intractable
at
first
glance
are
often
reduced
to
much
simpler
definite
integrals
by
performing
suitable
changes
of
coordinates
or
partial
integrations.
Thus:
Step
1:
Our
hypothetical
high-school
student
might
initially
be
tempted
to
perform
a
change
of
coordinates
e
−x
2
u
and
then
[erroneously!]
compute
∞
e
−x
2
−∞
dx
=
2
·
∞
−x
2
e
0
x=∞
dx
=
−
d(e
x=0
−x
2
1
)
=
du
=
1
0
—
only
to
realize
shortly
afterwards
that
this
computation
is
in
error,
on
account
of
the
erroneous
treatment
of
the
infinitesimal
“dx”
when
the
change
of
coordinates
was
executed.
Step
2:
This
realization
might
then
lead
the
student
to
attempt
to
repair
the
computation
of
Step
1
by
considering
various
iterated
partial
integrations
x=∞
x=∞
∞
1
1
−x
2
−x
2
−x
2
d(e
=
...
e
dx
=
−
)
=
e
d
2x
−∞
x=−∞
2x
x=−∞
—
which,
of
course,
lead
nowhere.
§
1.3.
Introduction
of
identical
but
mutually
alien
copies
At
this
point,
one
might
suggest
to
the
hypothetical
high-school
student
the
idea
of
computing
the
Gaussian
integral
by
first
squaring
the
integral
and
then
taking
the
square
root
of
the
value
of
the
square
of
the
integral.
That
is
to
say,
in
effect:
Step
3:
One
might
suggest
to
the
hypothetical
high-school
student
that
the
Gaussian
integral
can
in
fact
be
computed
by
considering
the
product
of
two
Shinichi
Mochizuki
8
identical
—
but
mutually
independent!
—
copies
of
the
Gaussian
integral
∞
∞
2
−x
2
e
dx
·
e
−y
dy
−∞
−∞
—
i.e.,
as
opposed
to
a
single
copy
of
the
Gaussian
integral.
Here,
let
us
recall
that
our
hypothetical
high-school
student
was
already
in
a
mental
state
of
extreme
frustration
as
a
result
of
the
student’s
intensive
and
heroic
attempts
in
Step
2
which
led
only
to
an
endless
labyrinth
of
meaningless
and
increasingly
complicated
mathematical
expressions.
This
experience
left
our
hypothetical
high-school
student
with
the
impression
that
the
Gaussian
integral
was
without
question
by
far
the
most
difficult
integral
that
the
student
had
ever
encountered.
In
light
of
this
experience,
the
suggestion
of
Step
3
evoked
a
reaction
of
intense
indignation
and
distrust
on
the
part
of
the
student.
That
is
to
say,
the
idea
that
meaningful
progress
could
be
made
in
the
computation
of
such
an
exceedingly
difficult
integral
simply
by
considering
two
identical
copies
of
the
integral
—
i.e.,
as
opposed
to
a
single
copy
—
struck
the
student
as
being
utterly
ludicrous.
Put
another
way,
the
suggestion
of
Step
3
was
simply
not
the
sort
of
suggestion
that
the
student
wanted
to
hear.
Rather,
the
student
was
keenly
interested
in
seeing
some
sort
of
clever
partial
integration
or
change
of
coordinates
involving
“sin(−)”,
“cos(−)”,
1
“tan(−)”,
“exp(−)”,
“
1+x
2
”,
etc.,
i.e.,
of
the
sort
that
the
student
was
used
to
seeing
in
familiar
expositions
of
one-variable
calculus.
§
1.4.
Integrals
over
two-dimensional
Euclidean
space
Only
after
quite
substantial
efforts
at
persuasion
did
our
hypothetical
high-school
student
reluctantly
agree
to
proceed
to
the
next
step
of
the
explanation:
Step
4:
If
one
considers
the
“totality”,
or
“total
space”,
of
the
coordinates
that
appear
in
the
product
of
two
copies
of
the
Gaussian
integral
of
Step
3,
then
one
can
regard
this
product
of
integrals
as
a
single
integral
2
2
−x
2
−y
2
e
·
e
dx
dy
=
e
−(x
+y
)
dx
dy
R
2
R
2
over
the
Euclidean
plane
R
2
.
Of
course,
our
hypothetical
high-school
student
might
have
some
trouble
with
Step
4
since
it
requires
one
to
assimilate
the
notion
of
an
integral
over
a
space,
i.e.,
the
Euclidean
plane
R
2
,
which
is
not
an
interval
of
the
real
line.
This,
however,
may
be
explained
by
reviewing
the
essential
philosophy
behind
the
notion
of
the
Riemann
integral
—
a
philosophy
which
should
be
familiar
from
one-variable
calculus:
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
9
Step
5:
One
may
think
of
integrals
over
more
general
spaces,
i.e.,
such
as
the
Euclidean
plane
R
2
,
as
computations
net
mass
=
lim
(infinitesimals
of
zero
mass)
of
“net
mass”
by
considering
limits
of
sums
of
infinitesimals,
i.e.,
such
as
“dx
dy”,
which
one
may
think
of
as
having
“zero
mass”.
§
1.5.
The
effect
on
infinitesimals
of
changes
of
coordinates
Just
as
in
one-variable
calculus,
computations
of
integrals
over
more
general
spaces
can
often
be
simplified
by
performing
suitable
changes
of
coordinates.
Any
[say,
con-
tinuously
differentiable]
change
of
coordinates
results
in
a
new
factor,
given
by
the
Jacobian,
in
the
integrand.
This
factor
constituted
by
the
Jacobian,
i.e.,
the
determi-
nant
of
a
certain
matrix
of
partial
derivatives,
may
appear
to
be
somewhat
mysterious
to
our
hypothetical
high-school
student,
who
is
only
familiar
with
changes
of
coordinates
in
one-variable
calculus.
On
the
other
hand,
the
appearance
of
the
Jacobian
may
be
justified
in
a
computational
fashion
as
follows:
Step
6:
Let
U,
V
⊆
R
2
be
open
subsets
of
R
2
and
U
(s,
t)
→
(x,
y)
=
(f
(s,
t),
g(s,
t))
∈
V
a
continuously
differentiable
change
of
coordinates
such
that
the
Jacobian
⎞
⎛
∂f
∂f
∂s
∂t
⎠
J
=
det
⎝
def
∂g
∂g
∂s
∂t
—
which
may
be
thought
of
as
a
continuous
real-valued
function
on
U
—
is
nonzero
throughout
U
.
Then
for
any
continuous
real-valued
functions
φ
:
U
→
R,
ψ
:
V
→
R
such
that
ψ(f
(s,
t),
g(s,
t))
=
φ(s,
t),
the
effect
of
the
above
change
of
coordinates
on
the
integral
of
ψ
over
V
may
be
computed
as
follows:
ψ
dx
dy
=
φ
·
J
ds
dt.
V
U
Step
7:
In
the
situation
of
Step
6,
the
effect
of
the
change
of
coordinates
on
the
“infinitesimals”
dx
dy
and
ds
dt
may
be
understood
as
follows:
First,
one
localizes
to
a
sufficiently
small
open
neighborhood
of
a
point
of
U
over
which
the
various
partial
derivatives
of
f
and
g
are
roughly
constant,
which
implies
that
the
change
of
coordinates
determined
by
f
and
g
is
roughly
linear.
Then
the
effect
of
such
a
linear
transformation
on
areas
—
i.e.,
in
the
language
of
Step
5,
“masses”
—
of
sufficiently
small
parallelograms
is
given
by
multiplying
Shinichi
Mochizuki
10
by
the
determinant
of
the
linear
transformation.
Indeed,
to
verify
this,
one
observes
that,
after
possible
pre-
and
post-composition
with
a
rotation
[which
clearly
does
not
affect
the
computation
of
such
areas],
one
may
assume
that
one
of
the
sides
of
the
parallelogram
under
consideration
is
a
line
segment
on
the
s-axis
whose
left-hand
endpoint
is
equal
to
the
origin
(0,
0),
and,
moreover,
that
the
linear
transformation
may
be
written
as
a
composite
of
toral
dilations
and
unipotent
linear
transformations
of
the
form
(s,
t)
→
(a
·
s,
b
·
t);
(s,
t)
→
(s
+
c
·
t,
t)
—
where
a,
b,
c
∈
R,
and
ab
=
0.
On
the
other
hand,
in
the
case
of
such
“upper
triangular”
linear
transformations,
the
effect
of
the
linear
transformation
on
the
area
of
the
parallelogram
under
consideration
is
an
easy
computation
at
the
level
of
high-school
planar
geometry.
§
1.6.
Passage
from
planar
cartesian
to
polar
coordinates
Once
the
“innocuous”
generalities
of
Steps
5,
6,
and
7
have
been
assimilated,
one
may
proceed
as
follows:
Step
8:
We
apply
Step
6
to
the
integral
of
Step
4,
regarded
as
an
integral
over
the
complement
R
2
\
(R
≤0
×
{0})
of
the
negative
x-axis
in
the
Euclidean
plane,
and
the
change
of
coordinates
R
>0
×
(−π,
π)
(r,
θ)
→
(x,
y)
=
(r
cos(θ),
r
sin(θ))
∈
R
2
\
(R
≤0
×
{0})
—
where
we
write
R
>0
for
the
set
of
positive
real
numbers
and
(−π,
π)
for
the
open
interval
of
real
numbers
between
−π
and
π.
Step
9:
The
change
of
coordinates
of
Step
8
allows
one
to
compute
as
follows:
∞
∞
2
2
−x
2
−y
2
e
dx
·
e
dy
=
e
−x
·
e
−y
dx
dy
2
−∞
−∞
R
2
2
=
e
−(x
+y
)
dx
dy
2
R
2
2
=
e
−(x
+y
)
dx
dy
R
2
\(R
≤0
×{0})
e
−r
rdr
dθ
2
=
R
>0
×(−π,π)
∞
e
=
0
−r
2
π
·
2rdr
·
−π
1
2
·
dθ
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
—
where
we
observe
that
the
final
equality
is
notable
in
that
it
shows
that,
in
the
computation
of
the
integral
under
consideration,
the
radial
[i.e.,
“r”]
and
angular
[i.e.,
“θ”]
coordinates
may
be
decoupled,
i.e.,
that
the
integral
under
consideration
may
be
written
as
a
product
of
a
radial
integral
and
an
angular
integral.
Step
10:
The
radial
integral
of
Step
9
may
be
evaluated
∞
e
−r
2
1
·
2rdr
=
d(e
0
−r
2
0
1
)
=
du
=
1
0
by
applying
the
change
of
coordinates
e
−r
2
u
that,
in
essence,
appeared
in
the
erroneous
initial
computation
of
Step
1!
Step
11:
The
angular
integral
of
Step
9
may
be
evaluated
as
follows:
π
1
2
·
dθ
=
π
−π
Here,
we
note
that,
if
one
thinks
of
the
Euclidean
plane
R
2
of
Step
4
as
the
complex
plane,
i.e.,
if
we
write
the
change
of
coordinates
of
Step
8
in
the
form
x
+
iy
=
r
·
e
iθ
,
then,
relative
to
the
Euclidean
coordinates
(x,
y)
of
Step
4,
the
above
evaluation
of
the
angular
integral
may
be
regarded
as
arising
from
the
change
of
coordinates
given
by
considering
the
imaginary
part
of
the
natural
logarithm
log(r
·
e
iθ
)
=
log(r)
+
iθ.
Step
12:
Thus,
in
summary,
we
conclude
that
∞
2
∞
∞
2
−x
2
−x
2
e
dx
=
e
dx
·
e
−y
dy
−∞
−∞
−∞
π
∞
−r
2
1
e
·
2rdr
·
=
π
=
2
·
dθ
0
−π
√
2
∞
π.
Here,
it
is
of
interest
to
observe
that,
—
i.e.,
that
−∞
e
−x
dx
=
although
in
the
approach
to
computing
the
Gaussian
integral
discussed
above
[i.e.,
starting
in
Step
3
and
concluding
in
the
present
Step
12],
the
radial
and
angular
integrals
of
Steps
10
and
11
arise
quite
naturally
in
the
final
compu-
tation
of
the
present
Step
12,
if
one
just
looks
at
the
original
Gaussian
integral
2
∞
e
−x
dx
on
the
real
line
from
a
naive
point
of
view
[cf.
Steps
1
and
2],
−∞
11
Shinichi
Mochizuki
12
it
is
essentially
a
hopeless
task
to
identify
“explicit
portions”
of
2
∞
this
original
Gaussian
integral
−∞
e
−x
dx
on
the
real
line
that
“correspond”
precisely,
in
some
sort
of
meaningful
sense,
to
the
radial
and
angular
integrals
of
Steps
10
and
11.
§
1.7.
Justification
of
naive
approach
up
to
an
“error
factor”
Put
another
way,
the
content
of
the
above
discussion
may
be
summarized
as
follows:
If
one
considers
two
identical
—
but
mutually
independent!
—
copies
of
the
Gaussian
integral,
i.e.,
as
opposed
to
a
single
copy,
then
the
naively
motivated
coordinate
transformation
that
gave
rise
to
the
erroneous
com-
√
putation
of
Step
1
may
be
“justified”,
up
to
a
suitable
“error
factor”
π!
In
this
context,
it
is
of
interest
to
note
that
the
technique
applied
in
the
above
discussion
2
for
evaluating
the
integral
of
the
Gaussian
distribution
“e
−x
”
cannot,
in
essence,
be
applied
to
integrals
of
functions
other
than
the
Gaussian
distribution.
Indeed,
this
essentially
unique
relationship
between
the
technique
of
the
above
discussion
and
the
Gaussian
distribution
may
be
understood
as
being,
in
essence,
a
consequence
of
the
fact
that
the
exponential
function
determines
an
isomorphism
of
Lie
groups
between
the
“additive
Lie
group”
of
real
numbers
and
the
“multiplicative
Lie
group”
of
positive
real
numbers.
We
refer
to
[Bell],
[Dawson]
for
more
details.
§
2.
§
2.1.
Changes
of
universe
as
arithmetic
changes
of
coordinates
The
issue
of
bounding
heights:
the
ABC
and
Szpiro
Conjectures
In
diophantine
geometry,
i.e.,
more
specifically,
the
diophantine
geometry
of
ra-
tional
points
of
an
algebraic
curve
over
a
number
field
[i.e.,
an
“NF”],
one
is
typically
concerned
with
the
issue
of
bounding
heights
of
such
rational
points.
A
brief
exposition
of
various
conjectures
related
to
this
issue
of
bounding
heights
of
ratio-
nal
points
may
be
found
in
[Fsk],
§1.3.
In
this
context,
the
case
where
the
algebraic
curve
under
consideration
is
the
projective
line
minus
three
points
corresponds
most
directly
to
the
so-called
ABC
and
—
by
thinking
of
this
projective
line
as
the
“λ-line”
that
appears
in
discussions
of
the
Legendre
form
of
the
Weierstrass
equation
for
an
elliptic
curve
—
Szpiro
Conjectures.
In
this
case,
the
height
of
a
rational
point
may
be
thought
of
as
a
suitable
weighted
sum
of
the
valuations
of
the
q-parameters
of
the
elliptic
curve
determined
by
the
rational
point
at
the
nonarchimedean
primes
of
po-
tentially
multiplicative
reduction
[cf.
the
discussion
at
the
end
of
[Fsk],
§2.2;
[GenEll],
Proposition
3.4].
Here,
it
is
also
useful
to
recall
[cf.
[GenEll],
Theorem
2.1]
that,
in
the
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
13
situation
of
the
ABC
or
Szpiro
Conjectures,
one
may
assume,
without
loss
of
generality,
that,
for
any
given
finite
set
Σ
of
[archimedean
and
nonarchimedean]
valuations
of
the
rational
number
field
Q,
the
rational
points
under
consideration
lie,
at
each
valuation
of
Σ,
inside
some
compact
subset
[i.e.,
of
the
set
of
rational
points
of
the
projective
line
minus
three
points
over
some
finite
extension
of
the
completion
of
Q
at
this
valuation]
satisfying
certain
properties.
In
particular,
when
one
computes
the
height
of
a
rational
point
of
the
projective
line
minus
three
points
as
a
suitable
weighted
sum
of
the
valuations
of
the
q-parameters
of
the
corresponding
elliptic
curve,
one
may
ignore,
up
to
bounded
discrepancies,
contri-
butions
to
the
height
that
arise,
say,
from
the
archimedean
valuations
or
from
the
nonarchimedean
valuations
that
lie
over
some
“exceptional”
prime
number
such
as
2.
§
2.2.
Arithmetic
degrees
as
global
integrals
As
is
well-known,
the
height
of
a
rational
point
may
be
thought
of
as
the
arith-
metic
degree
of
a
certain
arithmetic
line
bundle
over
the
field
of
definition
of
the
rational
point
[cf.
[Fsk],
§1.3;
[GenEll],
§1].
Alternatively,
from
an
idèlic
point
of
view,
such
arithmetic
degrees
of
arithmetic
line
bundles
over
an
NF
may
be
thought
of
as
logarithms
of
volumes
—
i.e.,
“log-volumes”
—
of
certain
regions
inside
the
ring
of
adèles
of
the
NF
[cf.
[Fsk],
§2.2;
[AbsTopIII],
Definition
5.9,
(iii);
[IUTchIII],
Proposi-
tion
3.9,
(iii)].
Relative
to
the
point
of
view
of
the
discussion
of
§1.4,
such
log-volumes
may
be
thought
of
as
“net
masses”,
that
is
to
say,
as
“global
masses”
[i.e.,
global
log-volumes]
that
arise
by
summing
up
various
“local
masses”
[i.e.,
local
log-volumes],
corresponding
to
the
[archimedean
and
nonarchimedean]
valuations
of
the
NF
under
consideration.
This
point
of
view
of
the
discussion
of
§1.4
suggests
further
that
such
a
global
net
mass
should
be
regarded
as
some
sort
of
integral
over
an
NF,
that
is
to
say,
which
arises
by
applying
some
sort
of
mysterious
“limit
summation
operation”
to
some
sort
of
“zero
mass
infinitesimal”
object
[i.e.,
corresponding
to
a
differential
form].
It
is
precisely
this
point
of
view
that
will
be
pursued
in
the
discussion
to
follow
via
the
following
correspondences
with
terminology
to
be
explained
below:
zero
mass
objects
←→
“étale-like”
structures
positive/nonzero
mass
objects
←→
“Frobenius-like”structures
Shinichi
Mochizuki
14
§
2.3.
Bounding
heights
via
global
multiplicative
subspaces
In
the
situation
discussed
in
§2.1,
one
way
to
understand
the
problem
of
showing
that
the
height
h
∈
R
of
a
rational
point
is
“small”
is
as
the
problem
of
showing
that,
for
some
fixed
natural
number
N
>
1,
the
height
h
satisfies
the
equation
def
N
·
h
=
h
+
h
+
...
+
h
=
h
[which
implies
that
h
=
0!]
—
or,
more
generally,
for
a
suitable
“relatively
small”
constant
C
∈
R
[i.e.,
which
is
independent
of
the
rational
point
under
consideration],
the
inequality
N
·
h
≤
h
+
C
[which
implies
that
h
≤
N
1
−1
·
C!]
—
holds.
Indeed,
this
is
precisely
the
approach
that
is
taken
to
bounding
heights
in
the
“tiny”
special
case
of
the
theory
of
[Falt1]
that
is
given
in
the
proof
of
[GenEll],
Lemma
3.5.
Here,
we
recall
that
the
key
assumption
in
[GenEll],
Lemma
3.5,
that
makes
this
sort
of
argument
work
is
the
assumption
of
the
existence,
for
some
prime
number
l,
of
a
certain
kind
of
special
rank
one
subspace
[i.e.,
a
subspace
whose
F
l
-dimension
is
equal
to
1]
of
the
space
of
l-torsion
points
[i.e.,
a
F
l
-
vector
space
of
dimension
2]
of
the
elliptic
curve
under
consideration.
Such
a
rank
one
subspace
is
typically
referred
to
in
this
context
as
a
global
multiplicative
subspace,
i.e.,
since
it
is
a
subspace
defined
over
the
NF
under
consideration
that
coincides,
at
each
nonarchimedean
valuation
of
the
NF
at
which
the
elliptic
curve
under
consideration
has
potentially
multiplicative
reduction,
with
the
rank
one
subspace
of
l-torsion
points
that
arises,
via
the
Tate
uniformization,
from
the
[one-dimensional]
space
of
l-torsion
points
of
the
multiplicative
group
G
m
.
The
quotient
of
the
original
given
elliptic
curve
by
such
a
global
multiplicative
subspace
is
an
elliptic
curve
that
is
isogenous
to
the
original
elliptic
curve.
Moreover,
the
q-parameters
of
this
isogenous
elliptic
curve
are
the
l-th
powers
of
the
q-parameters
of
the
original
elliptic
curve;
thus,
the
height
of
this
isogenous
elliptic
curve
is
[roughly,
up
to
contributions
of
negligible
order]
l
times
the
height
of
the
original
elliptic
curve.
These
properties
of
the
isogenous
elliptic
curve
allow
one
to
compute
the
height
of
the
isogenous
elliptic
curve
in
terms
of
the
height
of
the
original
elliptic
curve
by
calculating
the
effect
of
the
isogeny
relating
the
two
elliptic
curves
on
the
respective
sheaves
of
differentials
and
hence
to
conclude
an
inequality
“N
·
h
≤
h
+
C”
of
the
desired
type
[for
N
=
l
—
cf.
the
proof
of
[GenEll],
Lemma
3.5,
for
more
details].
At
a
more
concrete
level,
this
computation
may
be
summarized
as
the
observation
that,
by
considering
the
effect
of
the
isogeny
under
consideration
on
sheaves
of
differentials,
one
may
conclude
that
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
15
multiplying
heights
by
l
—
i.e.,
“raising
q-parameters
to
the
l-th
power”
q
→
q
l
—
has
the
effect
on
logarithmic
differential
forms
d
log(q)
=
dq
q
→
l
·
d
log(q)
of
multiplying
by
l,
i.e.,
at
the
level
of
heights,
of
adding
terms
of
the
order
of
log(l),
thus
giving
rise
to
inequalities
that
are
roughly
of
the
form
“l
·
h
≤
h
+
log(l)”.
On
the
other
hand,
in
general,
such
a
global
multiplicative
subspace
does
not
exist,
and
the
issue
of
somehow
“simulating”
the
existence
of
a
global
multiplicative
subspace
is
one
funda-
mental
theme
of
inter-universal
Teichmüller
theory.
§
2.4.
Bounding
heights
via
Frobenius
morphisms
on
number
fields
The
simulation
issue
discussed
in
§2.3
is,
in
some
sense,
the
fundamental
reason
for
the
construction
of
various
types
of
“Hodge
theaters”
in
[IUTchI]
[cf.
the
discussion
surrounding
[IUTchI],
Fig.
I1.4;
[IUTchI],
Remark
4.3.1].
From
the
point
of
view
of
the
present
discussion,
the
fundamental
additive
and
multiplicative
symmetries
that
appear
in
the
theory
of
[Θ
±ell
N
F
-]Hodge
theaters
[cf.
§3.3,
(v);
§3.6,
(i),
below]
and
which
correspond,
respectively,
to
the
additive
and
multiplicative
structures
of
the
ring
F
l
[where
l
is
the
fixed
prime
number
for
which
we
consider
l-torsion
points],
may
be
thought
of
as
corresponding,
respectively,
to
the
symmetries
in
the
equation
def
N
·
h
=
h
+
h
+
...
+
h
=
h
of
all
the
h’s
[in
the
case
of
the
additive
symmetry]
and
of
the
h’s
on
the
LHS
[in
the
case
of
the
multiplicative
symmetry].
This
portion
of
inter-universal
Teichmüller
theory
is
closely
related
to
the
analogy
between
inter-universal
Teichmüller
theory
and
the
classical
hyperbolic
geometry
of
the
upper
half-plane.
This
analogy
with
the
hyperbolic
geometry
of
the
upper
half-plane
is,
in
some
sense,
the
central
topic
of
[BogIUT]
[cf.
also
§3.10,
(vi);
§4.1,
(i);
§4.3,
(iii),
of
the
present
paper]
and
may
be
thought
of
as
corresponding
to
the
portion
of
inter-universal
Teichmüller
theory
discussed
in
[IUTchI],
[IUTchIII].
Since
this
aspect
of
inter-universal
Teichmüller
theory
is
already
discussed
in
substantial
detail
in
[BogIUT],
we
shall
not
discuss
it
in
much
detail
in
the
present
paper.
On
the
other
hand,
another
way
of
thinking
about
the
above
equation
“N
·
h
=
h”
is
as
follows:
This
equation
may
also
be
thought
of
as
calling
for
the
establishment
of
some
sort
of
analogue
for
an
NF
of
the
Frobenius
morphism
in
positive
character-
istic
scheme
theory,
i.e.,
a
Frobenius
morphism
that
somehow
“acts”
naturally
Shinichi
Mochizuki
16
on
the
entire
situation
[i.e.,
including
the
height
h,
as
well
as
the
q-parameters
at
nonarchimedean
valuations
of
potentially
multiplicative
reduction,
of
a
given
elliptic
curve
over
the
NF]
in
such
a
way
as
to
multiply
arithmetic
degrees
[such
as
the
height!]
by
N
and
raise
q-parameters
to
the
N
-th
power
—
i.e.,
h
→
N
·
h,
q
→
q
N
—
and
hence
yield
the
equation
“N
·
h
=
h”
[or
inequality
“N
·
h
≤
h
+
C”]
via
some
sort
of
natural
functoriality.
This
point
of
view
is
also
quite
fundamental
to
inter-universal
Teichmüller
theory,
and,
in
particular,
to
the
analogy
between
inter-universal
Teichmüller
theory
and
the
theory
of
the
Gaussian
integral,
as
reviewed
in
§1.
These
aspects
of
inter-universal
Teichmüller
theory
are
discussed
in
[IUTchII],
[IUTchIII].
In
the
present
paper,
we
shall
concentrate
mainly
on
the
exposition
of
these
aspects
of
inter-universal
Teichmüller
theory.
Before
proceeding,
we
remark
that,
ultimately,
in
inter-universal
Teichmüller
theory,
we
will,
in
effect,
take
“N
”
to
be
a
sort
of
symmetrized
average
over
the
def
squares
of
the
values
j
=
1,
2,
.
.
.
,
l
,
where
l
=
(l
−
1)/2,
and
l
is
the
prime
number
of
§2.3.
That
is
to
say,
whereas
the
[purely
hypothetical!]
naive
analogue
of
the
Frobenius
morphism
for
an
NF
considered
so
far
has
the
effect,
on
q-parameters
of
the
elliptic
curve
under
consideration
at
nonarchimedean
valuations
of
potentially
multiplicative
reduction,
of
mapping
q
→
q
N
,
the
sort
of
assignment
that
we
shall
ultimately
be
interested
in
in
inter-universal
Teichmüller
theory
is
an
assignment
[which
is
in
fact
typically
written
with
the
left-
and
right-
hand
sides
reversed]
2
q
→
{q
j
}
j=1,...,l
—
where
q
denotes
a
2l-th
root
of
the
q-parameter
q
—
i.e.,
an
assignment
which,
at
least
at
a
formal
level,
closely
resembles
a
Gaussian
distribution.
Of
course,
such
an
assignment
is
not
compatible
with
the
ring
structure
of
an
NF,
hence
does
not
exist
in
the
framework
of
conventional
scheme
theory.
Thus,
one
way
to
understand
inter-universal
Teichmüller
theory
is
as
follows:
in
some
sense
the
fundamental
theme
of
inter-universal
Teichmüller
theory
con-
sists
of
the
development
of
a
mechanism
for
computing
the
effect
—
e.g.,
on
heights
of
elliptic
curves
[cf.
the
discussion
of
§2.3!]
—
of
such
non-scheme-
theoretic
“Gaussian
Frobenius
morphisms”
on
NF’s.
§
2.5.
Fundamental
example
of
the
derivative
of
a
Frobenius
lifting
In
some
sense,
the
most
fundamental
example
of
the
sort
of
Frobenius
action
in
the
p-adic
theory
that
one
would
like
to
somehow
translate
into
the
case
of
NF’s
is
the
following
[cf.
[AbsTopII],
Remark
2.6.2;
[AbsTopIII],
§I5;
[IUTchIII],
Remark
3.12.4,
(v)]:
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
17
Example
2.5.1.
Frobenius
liftings
on
smooth
proper
curves.
Let
p
be
a
prime
number;
A
the
ring
of
Witt
vectors
of
a
perfect
field
k
of
characteristic
p;
X
a
smooth,
proper
curve
over
A
of
genus
g
X
≥
2;
Φ
:
X
→
X
a
Frobenius
lifting,
i.e.,
a
morphism
whose
reduction
modulo
p
coincides
with
the
Frobenius
morphism
in
characteristic
p.
Thus,
one
verifies
immediately
that
Φ
necessarily
lies
over
the
Frobenius
morphism
on
the
ring
of
Witt
vectors
A.
Write
ω
X
k
for
the
sheaf
of
differentials
of
def
X
k
=
X
×
A
k
over
k.
Then
the
derivative
of
Φ
yields,
upon
dividing
by
p,
a
morphism
of
line
bundles
Φ
∗
ω
X
k
→
ω
X
k
which
is
easily
verified
to
be
generically
injective.
Thus,
by
taking
global
degrees
of
line
bundles,
we
obtain
an
inequality
(p
−
1)(2g
X
−
2)
≤
0
—
hence,
in
particular,
an
inequality
g
X
≤
1
—
which
may
be
thought
of
as
being,
in
essence,
a
statement
to
the
effect
that
X
cannot
be
hyperbolic.
Note
that,
from
the
point
of
view
discussed
in
§1.4,
§1.5,
§2.2,
§2.3,
§2.4,
this
inequality
may
be
thought
of
as
a
computation
of
“global
net
masses”,
i.e.,
global
degrees
of
line
bundles
on
X
k
,
via
a
computation
of
the
effect
of
the
“change
of
coordinates”
Φ
by
considering
the
effect
of
this
change
of
coordinates
on
“infinitesimals”,
i.e.,
on
the
sheaf
of
differentials
ω
X
k
.
§
2.6.
Positive
characteristic
model
for
mono-anabelian
transport
One
fundamental
drawback
of
the
computation
discussed
in
Example
2.5.1
is
that
it
involves
the
operation
of
differentiation
on
X
k
,
an
operation
which
does
not,
at
least
in
the
immediate
literal
sense,
have
a
natural
analogue
in
the
case
of
NF’s.
This
drawback
does
not
exist
in
the
following
example,
which
treats
certain
subtle,
but
well-
known
aspects
of
anabelian
geometry
in
positive
characteristic
and,
moreover,
may,
in
some
sense,
be
regarded
as
the
fundamental
model,
or
prototype,
for
a
quite
substantial
portion
of
inter-universal
Teichmüller
theory.
In
this
example,
Galois
groups,
or
étale
fundamental
groups,
in
some
sense
play
the
role
that
is
played
by
tangent
bundles
in
the
classical
theory
—
a
situation
that
is
reminiscent
of
the
approach
of
the
[scheme-theoretic]
Hodge-Arakelov
theory
of
[HASurI],
[HASurII],
which
is
briefly
reviewed
in
§2.14
below.
One
notion
of
central
importance
in
this
example
—
and
indeed
throughout
inter-universal
Teichmüller
theory!
—
is
the
notion
of
a
cyclotome,
a
term
which
is
used
to
refer
to
an
isomorphic
copy
of
some
quotient
[by
a
closed
submodule]
of
the
familiar
Galois
module
“
Z(1)”,
i.e.,
the
“Tate
twist”
of
Shinichi
Mochizuki
18
the
trivial
Galois
module
“
Z”,
or,
alternatively,
the
rank
one
free
Z-module
equipped
with
the
action
determined
by
the
cyclotomic
character.
Also,
if
p
is
a
prime
number,
then
we
shall
write
Z
=
p
for
the
quotient
Z/Z
p
.
Example
2.6.1.
Mono-anabelian
transport
via
the
Frobenius
morphism
in
positive
characteristic.
(i)
Let
p
be
a
prime
number;
k
a
finite
field
of
characteristic
p;
X
a
smooth,
proper
curve
over
k
of
genus
g
X
≥
2;
K
the
function
field
of
X;
K
a
separable
closure
def
def
of
K.
Write
η
X
=
Spec(K);
η
X
=
Spec(
K);
k
⊆
K
for
the
algebraic
closure
of
k
Z
def
determined
by
K;
μ
k
⊆
k
for
the
group
of
roots
of
unity
of
k;
μ
k
=
p
=
Hom(Q/Z,
μ
k
);
def
def
G
K
=
Gal(
K/K);
G
k
=
Gal(k/k);
Π
X
for
the
quotient
of
G
K
determined
by
the
maximal
subextension
of
K
that
is
unramified
over
X;
Φ
X
:
X
→
X,
Φ
η
X
:
η
X
→
η
X
,
Φ
η
X
:
η
X
→
η
X
for
the
respective
Frobenius
morphisms
of
X,
η
X
,
η
X
.
Thus,
we
have
natural
sur-
jections
G
K
Π
X
G
k
,
and
Π
X
may
be
thought
of
as
[i.e.,
is
naturally
isomor-
phic
to]
the
étale
fundamental
group
of
X
[for
a
suitable
choice
of
basepoint].
Write
def
Δ
X
=
Ker(Π
X
G
k
).
Recall
that
it
follows
from
elementary
facts
concerning
sep-
arable
and
purely
inseparable
field
extensions
that
[by
considering
Φ
η
X
]
Φ
η
X
induces
isomorphisms
of
Galois
groups
and
étale
fundamental
groups
∼
Ψ
X
:
Π
X
→
Π
X
,
∼
Ψ
η
X
:
G
K
→
G
K
—
which
is,
in
some
sense,
a
quite
remarkable
fact
since
the
Frobenius
morphisms
Φ
X
,
Φ
η
X
themselves
are
morphisms
“of
degree
p
>
1”,
hence,
in
particular,
are
by
no
means
isomorphisms!
We
refer
to
[IUTchIV],
Example
3.6,
for
a
more
general
version
of
this
phenomenon.
(ii)
Next,
let
us
recall
that
it
follows
from
the
fundamental
anabelian
results
of
[Uchi]
that
there
exists
a
purely
group-theoretic
functorial
algorithm
G
K
→
K(G
K
)
CFT
G
K
—
i.e.,
an
algorithm
whose
input
data
is
the
abstract
topological
group
G
K
,
whose
functoriality
is
with
respect
to
isomorphisms
of
topological
groups,
and
whose
output
data
is
a
field
K(G
K
)
CFT
equipped
with
a
G
K
-action.
Moreover,
if
one
allows
oneself
to
apply
the
conventional
interpretation
of
G
K
as
a
Galois
group
Gal(
K/K),
then
there
is
a
natural
G
K
-equivariant
isomorphism
∼
ρ
:
K
→
K(G
K
)
CFT
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
19
that
arises
from
the
reciprocity
map
of
class
field
theory,
applied
to
each
of
the
finite
subextensions
of
the
extension
K/K.
Since
class
field
theory
depends,
in
an
essential
way,
on
the
field
structure
of
K
and
K,
it
follows
formally
that,
at
least
in
an
a
priori
sense,
the
construction
of
ρ
itself
also
depends,
in
an
essential
way,
on
the
field
structure
of
K
and
K.
Moreover,
the
fact
that
the
isomorphism
K(Ψ
η
X
)
CFT
:
∼
K(G
K
)
CFT
→
K(G
K
)
CFT
and
ρ
are
[unlike
Φ
η
X
itself!]
isomorphisms
implies
that
the
diagram
η
)
CFT
K(Ψ
X
←−
K(G
K
)
CFT
K(G
K
)
CFT
⏐
ρ
⏐
⏐
ρ
⏐
?
Φ
∗
η
X
−→
K
K
fails
to
be
commutative!
(iii)
On
the
other
hand,
let
us
recall
that
consideration
of
the
first
Chern
class
of
a
line
bundle
of
degree
1
on
X
yields
a
natural
isomorphism
Z
∼
def
λ
:
μ
k
=
p
→
M
X
=
Hom
Z
=
p
(H
2
(Δ
X
,
Z
=
p
),
Z
=
p
)
[cf.,
e.g.,
[Cusp],
Proposition
1.2,
(ii)].
Such
a
natural
isomorphism
between
cyclotomes
Z
[i.e.,
such
as
μ
k
=
p
,
M
X
]
will
be
referred
to
as
a
cyclotomic
rigidity
isomorphism.
Thus,
if
we
let
“H”
range
over
the
open
subgroups
of
G
K
,
then,
by
composing
this
cyclotomic
rigidity
isomorphism
[applied
to
the
coefficients
of
“H
1
(−)”]
with
the
Kummer
mor-
phism
associated
to
the
multiplicative
group
(
K
H
)
×
of
the
field
K
H
of
H-invariants
of
K,
we
obtain
an
embedding
Z
∼
κ
:
K
×
→
lim
H
1
(H,
μ
k
=
p
)
→
lim
H
1
(H,
M
X
)
−→
−→
H
H
—
whose
construction
depends
only
on
the
multiplicative
monoid
with
G
K
-action
K
×
and
the
cyclotomic
rigidity
isomorphism
λ.
Note
that
the
existence
of
the
re-
construction
algorithm
K(−)
CFT
reviewed
above
implies
that
the
kernels
of
the
natural
surjections
G
K
Π
X
G
k
may
be
reconstructed
group-theoretically
from
the
ab-
stract
topological
group
G
K
.
In
particular,
we
conclude
that
lim
H
H
1
(H,
M
X
)
may
be
−→
reconstructed
group-theoretically
from
the
abstract
topological
group
G
K
.
Moreover,
the
anabelian
theory
of
[Cusp]
[cf.,
especially,
[Cusp],
Proposition
2.1;
[Cusp],
Theorem
2.1,
(ii);
[Cusp],
Theorem
3.2]
yields
a
purely
group-theoretic
functorial
algorithm
G
K
→
K(G
K
)
Kum
G
K
—
i.e.,
an
algorithm
whose
input
data
is
the
abstract
topological
group
G
K
,
whose
functoriality
is
with
respect
to
isomorphisms
of
topological
groups,
and
whose
output
data
is
a
field
K(G
K
)
Kum
equipped
with
a
G
K
-action
which
is
constructed
as
the
union
Shinichi
Mochizuki
20
with
{0}
of
the
image
of
κ.
[In
fact,
the
input
data
for
this
algorithm
may
be
taken
to
be
the
abstract
topological
group
Π
X
,
but
we
shall
not
pursue
this
topic
here.]
Thus,
just
as
in
the
case
of
“
K(−)
CFT
”,
the
fact
that
the
isomorphism
K(Ψ
η
X
)
Kum
:
∼
K(G
K
)
Kum
→
K(G
K
)
Kum
and
κ
are
[unlike
Φ
η
X
itself!]
isomorphisms
implies
that
the
diagram
η
)
Kum
K(Ψ
X
Kum
←−
K(G
K
)
Kum
K(G
K
)
⏐
κ
⏐
K
?
Φ
∗
η
X
−→
⏐
κ
⏐
K
—
where,
by
a
slight
abuse
of
notation,
we
write
“κ”
for
the
“formal
union”
of
κ
with
“{0}”
—
fails
to
be
commutative!
(iv)
The
[a
priori]
noncommutativity
of
the
diagram
of
the
final
display
of
(iii)
may
be
interpreted
in
two
ways,
as
follows:
(a)
If
one
starts
with
the
assumption
that
this
diagram
is
in
fact
commutative,
then
the
fact
that
the
Frobenius
morphism
Φ
∗
η
X
multiplies
degrees
of
rational
functions
∈
K
by
p,
together
with
the
fact
that
the
vertical
and
upper
horizontal
arrows
of
the
diagram
are
isomorphisms,
imply
[since
the
field
K
is
not
perfect!]
the
erroneous
conclusion
that
all
degrees
of
rational
functions
∈
K
are
equal
to
zero!
This
sort
of
argument
is
formally
similar
to
the
argument
“N
·
h
=
h
=⇒
h
=
0”
discussed
in
§2.3.
(b)
One
may
regard
the
noncommutativity
of
this
diagram
as
the
problem
of
com-
puting
just
how
much
“indeterminacy”
one
must
allow
in
the
objects
and
arrows
that
appear
in
the
diagram
in
order
to
render
the
diagram
commutative.
From
this
point
of
view,
one
verifies
immediately
that
a
solution
to
this
problem
may
be
given
by
introducing
“indeterminacies”
as
follows:
One
replaces
λ
λ
·
p
Z
the
cyclotomic
rigidity
isomorphism
λ
by
the
orbit
of
λ
with
respect
to
com-
position
with
multiplication
by
arbitrary
Z-powers
of
p,
and
one
replaces
K
K
pf
,
K(G
K
)
Kum
(
K(G
K
)
Kum
)
pf
the
fields
K,
K(G
K
)
Kum
by
their
perfections.
Here,
we
observe
that
intepretation
(b)
may
be
regarded
as
corresponding
to
the
argu-
ment
“N
·
h
≤
h
+
C
=⇒
h
≤
N
1
−1
·
C”
discussed
in
§2.3.
That
is
to
say,
If,
in
the
situation
of
(b),
one
can
show
that
the
indeterminacies
necessary
to
render
the
diagram
commutative
are
sufficiently
mild,
at
least
in
the
case
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
21
of
the
heights
or
q-parameters
that
one
is
interested
in,
then
it
is
“reasonable
to
expect”
that
the
resulting
“contradiction
in
the
style
of
interpretation
(a)”
between
multiplying
degrees
by
some
integer
[or
rational
number]
>
1
and
the
fact
that
the
vertical
and
upper
horizontal
arrows
of
the
diagram
are
isomorphisms
should
enable
one
to
conclude
that
“N
·
h
≤
h
+
C”
[and
hence
that
“h
≤
1
N
−1
·
C”].
This
is
precisely
the
approach
that
is
in
fact
taken
in
inter-universal
Teichmüller
theory.
§
2.7.
The
apparatus
and
terminology
of
mono-anabelian
transport
Example
2.6.1
is
exceptionally
rich
in
structural
similarities
to
inter-universal
Teichmüller
theory,
which
we
proceed
to
explain
in
detail
as
follows.
One
way
to
un-
derstand
these
structural
similarities
is
by
considering
the
quite
substantial
portion
of
terminology
of
inter-universal
Teichmüller
theory
that
was,
in
essence,
inspired
by
Example
2.6.1:
(i)
Links
between
“mutually
alien”
copies
of
scheme
theory:
One
central
aspect
of
inter-universal
Teichmüller
theory
is
the
study
of
certain
“walls”,
or
“filters”
—
which
are
often
referred
to
as
“links”
—
that
separate
two
“mutually
alien”
copies
of
conventional
scheme
theory
[cf.
the
discussions
of
[IUTchII],
Remark
3.6.2;
[IUTchIV],
Remark
3.6.1].
The
main
example
of
such
a
link
in
inter-universal
Teichmüller
theory
is
constituted
by
[various
versions
of]
the
Θ-link.
The
log-link
also
plays
an
important
role
in
inter-universal
Teichmüller
theory.
The
main
motivating
example
for
these
links
which
play
a
central
role
in
inter-universal
Teichmüller
theory
is
the
Frobenius
morphism
Φ
η
X
of
Example
2.6.1.
From
the
point
of
view
of
the
discussion
of
§1.4,
§1.5,
§2.2,
§2.3,
§2.4,
and
§2.5,
such
a
link
corresponds
to
a
change
of
coordinates.
(ii)
Frobenius-like
objects:
The
objects
that
appear
on
either
side
of
a
link
and
which
are
used
in
order
to
construct,
or
“set
up”,
the
link,
are
referred
to
as
“Frobenius-like”.
Put
another
way,
Frobenius-like
objects
are
objects
that,
at
least
a
priori,
are
only
defined
on
one
side
of
a
link
[i.e.,
either
the
domain
or
codomain],
and,
in
particular,
do
not
necessarily
map
isomorphically
to
corresponding
objects
on
the
opposite
side
of
the
link.
22
Shinichi
Mochizuki
Thus,
in
Example
2.6.1,
the
“mutually
alien”
copies
of
K
on
either
side
of
the
p-
power
map
Φ
∗
η
X
are
Frobenius-like.
Typically,
Frobenius-like
structures
are
characterized
by
the
fact
that
they
have
positive/nonzero
mass.
That
is
to
say,
Frobenius-like
structures
represent
the
positive
mass
—
i.e.,
such
as
degrees
of
rational
functions
in
Example
2.6.1
or
heights/degrees
of
arithmetic
line
bundles
in
the
context
of
diophantine
geometry
—
that
one
is
ultimately
interested
in
computing
and,
moreover,
is,
at
least
in
an
a
priori
sense,
affected
in
a
nontrivial
way,
e.g.,
multiplied
by
some
factor
>
1,
by
the
link
under
consideration.
From
this
point
of
view,
Frobenius-like
objects
are
characterized
by
the
fact
that
the
link
under
consideration
gives
rise
to
an
“ordering”,
or
“asymmetry”,
between
Frobenius-like
objects
in
the
domain
and
codomain
of
the
link
under
consideration
[cf.
the
discussion
of
[FrdI],
§I3,
§I4].
(iii)
Étale-like
objects:
By
contrast,
objects
that
appear
on
either
side
of
a
link
that
correspond
to
the
“topology
of
some
sort
of
underlying
space”
—
such
as
the
étale
topology!
—
are
referred
to
as
“étale-like”.
Typically,
étale-like
structures
are
mapped
isomorphically
—
albeit
via
some
indeterminate
isomorphism!
[cf.
the
discussion
of
§2.10
below]
—
to
one
another
by
the
link
under
consideration.
From
this
point
of
view,
étale-like
objects
are
characterized
by
the
fact
that
the
link
under
consideration
gives
rise
to
a
“confusion”,
or
“symmetry”,
between
étale-like
objects
in
the
domain
and
codomain
of
the
link
under
consideration
[cf.
the
discussion
of
[FrdI],
§I3,
§I4].
Thus,
in
Example
2.6.1,
the
Galois
groups/étale
fundamental
groups
G
K
,
Π
X
,
which
are
mapped
isomorphically
to
one
another
via
Φ
η
X
,
albeit
via
some
“mysterious
indeter-
minate
isomorphism”,
are
étale-like.
Objects
that
are
algorithmically
constructed
from
étale-like
objects
such
as
G
K
or
Π
X
are
also
referred
to
as
étale-like,
so
long
as
they
are
regarded
as
being
equipped
with
the
additional
structure
constituted
by
the
algorithm
applied
to
construct
the
object
from
some
object
such
as
G
K
or
Π
X
.
Étale-like
structures
are
regarded
as
having
zero
mass
and
are
used
as
rigid
contain-
ers
for
positive
mass
Frobenius-like
objects,
i.e.,
containers
whose
structure
satisfies
certain
rigidity
properties
that
typically
arise
from
various
anabelian
properties
and
[as
in
Example
2.6.1!]
allows
one
to
compute
the
effect
on
positive
mass
Frobenius-like
objects
of
the
links,
or
“changes
of
coordinates”,
under
consideration.
(iv)
Coric
objects:
In
the
context
of
consideration
of
some
sort
of
link
as
in
(i),
coricity
refers
to
the
property
of
being
invariant
with
respect
to
—
i.e.,
the
property
of
mapping
isomorphically
to
a
corresponding
object
on
the
opposite
side
of
—
the
link
under
consideration.
Thus,
as
discussed
in
(iii),
étale-like
objects,
considered
up
to
isomorphism,
constitute
a
primary
example
of
the
notion
of
a
coric
object.
On
the
other
hand,
[non-étale-like]
Frobenius-like
coric
objects
also
arise
naturally
in
various
contexts.
Indeed,
in
the
situation
of
Example
2.6.1
[cf.
the
discussion
of
Example
2.6.1,
(iv),
(b)],
not
only
étale-like
objects
such
as
Π
X
,
G
K
,
and
K(G
K
)
Kum
,
but
also
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
étale-like
object
Kummer
⇑
étale-like
object
mono-anabelian
rigidity
isom.
Frobenius-like
object
Kummer
link
23
⇓
isom.
−1
Frobenius-like
object
Fig.
2.1:
Mono-anabelian
transport
Frobenius-like
objects
such
as
the
perfections
K
pf
are
coric.
(v)
The
computational
technique
of
mono-anabelian
transport:
The
tech-
nique
discussed
in
(iii),
i.e.,
of
computing
the
effect
on
positive
mass
Frobenius-like
objects
of
the
links,
or
“changes
of
coordinates”,
under
consideration,
by
first
applying
some
sort
of
Kummer
isomorphism
to
pass
from
Frobenius-like
to
corresponding
étale-
like
objects,
then
applying
some
sort
of
anabelian
construction
algorithm
or
rigidity
property,
and
finally
applying
a
suitable
inverse
Kummer
isomor-
phism
to
pass
from
étale-like
to
corresponding
Frobenius-like
objects
will
be
referred
to
as
the
technique
of
mono-anabelian
transport
[cf.
Fig.
2.1
above].
In
some
sense,
the
most
fundamental
prototype
for
this
technique
is
the
situation
described
in
Example
2.6.1,
(iii)
[cf.
also
the
discussion
of
Example
2.6.1,
(iv),
(b)].
Here,
the
term
“mono-anabelian”
[cf.
the
discussion
of
[AbsTopIII],
§I2]
refers
to
the
fact
that
the
algorithm
under
consideration
is
an
algorithm
whose
input
data
[typically]
consists
only
of
an
abstract
profinite
group
[i.e.,
that
“just
happens”
to
be
isomorphic
to
a
Galois
group
or
étale
fundamental
group
that
arises
from
scheme
theory!].
This
term
is
used
to
distinguish
from
fully
faithfulness
results
[i.e.,
to
the
effect
that
one
has
a
natural
bijection
between
certain
types
of
morphisms
of
schemes
and
certain
types
of
morphisms
of
profinite
groups]
of
the
sort
that
appear
in
various
anabelian
conjectures
of
Grothendieck.
Such
fully
faithfulness
results
are
referred
to
as
“bi-anabelian”.
(vi)
Kummer-detachment
indeterminacies
versus
étale-transport
indeter-
minacies:
The
first
step
that
occurs
in
the
procedure
for
mono-anabelian
transport
[cf.
the
discussion
of
(v)]
is
the
passage,
via
some
sort
of
Kummer
isomorphism,
from
Frobenius-like
objects
to
corresponding
étale-like
objects.
This
first
step
is
referred
to
as
Kummer-detachment.
The
indeterminacies
that
arise
during
this
first
step
are
re-
ferred
to
as
Kummer-detachment
indeterminacies
[cf.
[IUTchIII],
Remark
1.5.4].
Shinichi
Mochizuki
24
Such
Kummer-detachment
indeterminacies
typically
involve
indeterminacies
in
the
cy-
clotomic
rigidity
isomorphism
that
is
applied,
i.e.,
as
in
the
situation
discussed
in
Example
2.6.1,
(iv),
(b).
On
the
other
hand,
in
general,
more
complicated
Kummer-
detachment
indeterminacies
[i.e.,
that
are
not
directly
related
to
cyclotomic
rigidity
isomorphisms]
can
occur.
By
contrast,
the
indeterminacies
that
occur
as
a
result
of
the
fact
that
the
étale-like
structures
under
consideration
may
only
be
regarded
as
being
known
up
to
an
indeterminate
isomorphism
[cf.
the
discussion
of
(iii),
as
well
as
of
§2.10
below]
are
referred
to
as
étale-transport
indeterminacies
[cf.
[IUTchIII],
Remark
1.5.4].
(vii)
Arithmetic
holomorphic
structures
versus
mono-analytic
structures:
A
ring
may
be
regarded
as
consisting
of
“two
combinatorial
dimensions”
—
namely,
the
underlying
additive
and
multiplicative
structures
of
the
ring
—
which
are
in-
tertwined
with
one
another
in
a
rather
complicated
fashion
[cf.
the
discussion
of
[AbsTopIII],
§I3;
[AbsTopIII],
Remark
5.6.1].
In
inter-universal
Teichmüller
theory,
one
is
interested
in
dismantling
this
complicated
intertwining
structure
by
considering
the
underlying
additive
and
multiplicative
monoids
associated
to
a
ring
separately.
In
this
context,
the
ring
structure,
as
well
as
other
structures
such
as
étale
fundamen-
tal
groups
that
are
sufficiently
rigid
as
to
allow
the
algorithmic
reconstruction
of
the
ring
structure,
are
referred
to
as
arithmetic
holomorphic
structures.
By
contrast,
structures
that
arise
from
dismantling
the
complicated
intertwining
inherent
in
a
ring
structure
are
referred
to
as
mono-analytic
—
a
term
which
may
be
thought
of
as
a
sort
of
arithmetic
analogue
of
the
notion
of
an
underlying
real
analytic
structure
in
the
context
of
complex
holomorphic
structures.
From
this
point
of
view,
the
approach
of
Example
2.6.1,
(ii),
involving
the
reciprocity
map
of
class
field
theory
depends
on
the
arithmetic
holomorphic
structure
[i.e.,
the
ring
structure]
of
the
field
K
in
a
quite
essential
and
complicated
way.
By
contrast,
the
Kummer-theoretic
approach
of
Example
2.6.1,
(iii),
only
depends
on
the
mono-analytic
structure
constituted
by
the
underlying
multiplicative
monoid
of
the
field
K,
together
with
the
cyclotomic
rigidity
isomorphism
λ.
Thus,
although
λ
depends
on
the
ring
structure
of
the
field
K,
the
Kummer-theoretic
approach
of
Example
2.6.1,
(iii),
has
the
advantage,
from
the
point
of
view
of
dismantling
the
arithmetic
holomorphic
structure,
of
isolating
the
dependence
of
κ
on
the
arithmetic
holomorphic
structure
of
the
field
K
in
the
“compact
form”
constituted
by
the
cyclotomic
rigidity
isomor-
phism
λ.
§
2.8.
Remark
on
the
usage
of
certain
terminology
In
the
context
of
the
discussion
of
§2.7,
we
remark
that
although
terms
such
as
“link”,
“Frobenius-like”,
“étale-like”,
“coric”,
“mono-anabelian
transport”,
“Kummer-
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
25
detachment”,
“cyclotomic
rigidity
isomorphism”,
“mono-analytic”,
and
“arithmetic
holo-
morphic
structure”
are
well-defined
in
the
various
specific
contexts
in
which
they
are
applied,
these
terms
do
not
admit
general
definitions
that
are
applicable
in
all
contexts.
In
this
sense,
such
terms
are
used
in
a
way
that
is
similar
to
the
way
in
which
terms
such
as
“underlying”
[cf.,
e.g.,
the
“underlying
topological
space
of
a
scheme”,
the
“underlying
real
analytic
manifold
of
a
complex
manifold”]
or
“anabelian”
are
typically
used
in
mathematical
discussions.
The
term
“multiradial”,
which
will
be
discussed
in
§3,
is
also
used
in
this
way.
In
this
context,
we
remark
that
one
aspect
that
complicates
the
use
of
the
terms
“Frobenius-like”
and
“étale-like”
is
the
sort
of
curious
process
of
evolution
that
these
terms
underwent
as
the
author
progressed
from
writing
[FrdI],
[FrdII]
in
2005
to
writing
[IUTchI]
in
2008
and
finally
to
writing
[IUTchII]
and
[IUTchIII]
during
the
years
2009
-
2010.
This
“curious
process
of
evolution”
may
be
summarized
as
follows:
(1
Fr/ét
(2
Fr/ét
)
Vague
philosophical
approach
[i.e.,
“order-conscious”
vs.
“indif-
ferent
to
order”]:
The
first
stage
in
this
evolutionary
process
consists
of
the
vague
philosophical
characterizations
of
the
notion
of
“Frobenius-like”
via
the
term
“order-conscious”
and
of
the
notion
of
“étale-like”
via
the
phrase
“indifferent
to
order”.
These
vague
characterizations
were
motivated
by
the
situation
surround-
ing
the
monoids
[i.e.,
in
the
case
of
“Frobenius-like”]
that
appear
in
the
the-
ory
of
Frobenioids
[cf.
[FrdI],
[FrdII]]
and
the
situation
surrounding
the
Galois
groups/arithmetic
fundamental
groups
[i.e.,
in
the
case
of
“étale-like”]
that
appear
in
the
base
categories
[“D”]
of
Frobenioids.
This
point
of
view
is
discussed
in
[FrdI],
§I4,
and
is
quoted
and
applied
throughout
[IUTchI].
)
Characterization
in
the
context
of
the
log-theta-lattice:
This
point
of
view
consists
of
the
characterization
of
the
notions
of
“Frobenius-like”
and
“étale-
like”
in
the
context
of
the
specific
links,
i.e.,
the
Θ-
and
log-links,
that
occur
in
the
log-theta-lattice.
This
approach
is
developed
throughout
[IUTchII]
[cf.,
especially,
[IUTchII],
Remark
3.6.2]
and
[IUTchIII]
[cf.,
especially,
[IUTchIII],
Remark
1.5.4].
Related
discussions
may
be
found
in
[IUTchIV],
Remarks
3.6.2,
3.6.3.
At
a
purely
technical/notational
level,
this
approach
may
be
understood
as
follows
[cf.
also
the
discussion
in
the
final
portion
of
§3.3,
(vii),
of
the
present
paper]:
·
Frobenius-like
objects
[or
structures]
are
objects
that,
when
embedded
in
the
log-
theta-lattice,
are
marked
[via
left-hand
superscripts]
by
lattice
coordinates
“(n,
m)”
or,
when
not
embedded
in
the
log-theta-lattice,
are
marked
[via
left-hand
super-
scripts]
by
daggers
“†”/double
daggers
“‡”/asterisks
“∗”.
Shinichi
Mochizuki
26
·
étale-like
objects
[or
structures]
are
objects
that
arise
from
[i.e.,
are
often
de-
noted
as
“functions
(−)”
of]
the
D/D
-portions
of
the
Θ
±ell
N
F
-Hodge
theaters
that
appear,
or,
when
embedded
in
the
log-theta-lattice,
are
marked
[via
left-hand
superscripts]
by
vertically
coric/bi-coric
lattice
coordinates
“(n,
◦)”/“(◦,
◦)”.
(3
Fr/ét
)
Abstract
link-theoretic
approach:
This
is
the
the
approach
taken
in
§2.7
of
the
present
paper.
In
this
approach,
the
notions
of
“Frobenius-like”
and
“étale-like”
are
only
defined
in
the
context
of
a
specific
link.
This
approach
arose,
in
discus-
sions
involving
the
author
and
Y.
Hoshi,
as
a
sort
of
abstraction/generalization
of
the
situation
that
occurs
in
[IUTchII],
[IUTchIII]
in
the
case
of
the
specific
links,
Fr/ét
)].
i.e.,
the
Θ-
and
log-links,
that
occur
in
the
case
of
the
log-theta-lattice
[cf.
(2
Fr/ét
Fr/ét
Fr/ét
Fr/ét
Thus,
of
these
three
approaches
(1
),
(2
),
(3
),
the
approach
(3
)
is,
in
some
sense,
theoretically
the
most
satisfying
approach,
especially
from
the
point
of
view
of
considering
possible
generalizations
of
the
theory
of
[IUTchI],
[IUTchII],
[IUTchIII],
[IUTchIV].
On
the
other
hand,
from
the
point
of
view
of
the
more
restricted
goal
of
attaining
a
technically
sound
understanding
of
the
content
of
[IUTchI],
[IUTchII],
Fr/ét
)
is
[IUTchIII],
[IUTchIV],
the
purely
technical/notational
approach
discussed
in
(2
quite
sufficient.
§
2.9.
Mono-anabelian
transport
and
the
Kodaira-Spencer
morphism
The
discussion
of
§2.6
and
§2.7
may
be
summarized
as
follows:
In
some
sense,
the
central
theme
of
inter-universal
Teichmüller
theory
is
the
computation
via
mono-
anabelian
transport
—
in
the
spirit
of
the
discussion
of
Example
2.6.1,
(iv),
(b)
—
of
the
discrepancy
between
two
[systems
of]
Kummer
theories,
that
is
to
say,
of
the
sort
of
indeterminacies
that
one
must
admit
in
order
to
render
two
systems
of
Kummer
theories
compatible
—
i.e.,
relative
to
the
various
gluings
constituted
by
some
link
[cf.
the
Frobenius
morphism
Φ
η
X
in
Example
2.6.1;
the
discussion
of
§2.7,
(i)]
between
the
two
systems
of
Kummer
theories
—
with
simultaneous
execution,
e.g.,
when
one
of
the
two
systems
of
Kummer
theories
[cf.
the
objects
in
the
lower
right-hand
corner
of
the
diagram
of
§2.6,
(iii),
and
Fig.
2.1]
is
held
fixed.
In
this
context,
it
is
of
interest
to
observe
that
this
approach
of
computing
degrees
of
[“positive
mass”]
Frobenius-like
objects
by
embedding
them
into
rigid
étale-like
[“zero
mass”]
containers
[cf.
the
discussion
of
§2.7,
(iii),
(v)]
is
formally
similar
to
the
classical
definition
of
the
Kodaira-Spencer
morphism
associated
to
a
family
of
elliptic
curves
[cf.
also
the
discussion
of
§3.1,
(v),
below]:
Indeed,
suppose
[relative
to
the
terminology
of
[Semi],
§0]
that
S
log
is
a
smooth
log
curve
over
Spec(C)
[equipped
with
the
trivial
log
structure],
and
that
E
log
→
S
log
is
a
stable
log
curve
of
type
(1,
1)
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
27
[i.e.,
in
essence,
a
family
of
elliptic
curves
whose
origin
is
regarded
as
a
“marked
point”,
and
which
is
assumed
to
have
stable
reduction
at
the
points
of
degeneration].
Write
ω
S
log
/C
for
the
sheaf
of
relative
logarithmic
differentials
of
S
log
→
Spec(C);
(E,
∇
E
:
E
→
E
⊗
O
S
ω
S
log
/C
)
for
the
rank
two
vector
bundle
with
logarithmic
connection
on
S
log
determined
by
the
first
logarithmic
de
Rham
cohomology
module
of
E
log
→
S
log
and
the
logarithmic
Gauss-Manin
connection;
ω
E
⊆
E
for
the
Hodge
filtration
on
E;
τ
E
for
the
O
S
-dual
of
ω
E
.
Thus,
we
have
a
natural
exact
sequence
0
→
ω
E
→
E
→
τ
E
→
0.
Then
the
Kodaira-Spencer
morphism
associated
to
the
family
E
log
→
S
log
may
be
defined
as
the
composite
of
morphisms
in
the
diagram
∇
E
E
⊗
O
S
ω
S
log
/C
E
−→
⏐
⏐
⏐
⏐
ω
E
τ
E
⊗
O
S
ω
S
log
/C
—
where
the
left-hand
vertical
arrow
is
the
natural
inclusion,
and
the
right-hand
verti-
cal
arrow
is
the
natural
surjection.
The
three
arrows
in
this
diagram
may
be
regarded
as
corresponding
to
the
three
arrows
in
analogous
positions
in
the
diagrams
of
Example
2.6.1,
(iii),
and
Fig.
2.1.
That
is
to
say,
this
diagram
may
be
understood
as
a
compu-
tation
of
the
degree
deg(−)
of
the
“positive
mass”
Frobenius-like
object
ω
E
that
yields
an
inequality
deg(ω
E
)
≤
deg(ω
S
log
/C
)
−
deg(ω
E
),
i.e.,
2
·
deg(ω
E
)
≤
deg(ω
S
log
/C
)
via
a
comparison
of
“alien
copies”
of
the
“positive
mass”
Frobenius-like
object
ω
E
that
lie
on
opposite
sides
of
the
“link”
constituted
by
a
deformation
of
the
mod-
uli/holomorphic
structure
of
the
family
of
elliptic
curves
under
consideration.
This
comparison
is
performed
by
relating
these
Frobenius-like
objects
ω
E
[or
its
dual]
on
either
side
of
the
deformation
by
means
of
the
“zero
mass”
étale-like
object
(E,
∇
E
),
i.e.,
which
may
be
thought
of
as
a
local
system
on
the
open
subscheme
U
S
⊆
S
of
the
underlying
scheme
S
of
S
log
where
the
log
structure
of
S
log
is
trivial.
The
tensor
product
with
ω
S
log
/C
—
and
the
resulting
appearance
of
the
bound
deg(ω
S
log
/C
)
in
the
above
inequality
—
may
be
understood
as
the
indeterminacy
that
one
must
admit
in
order
to
achieve
this
comparison.
§
2.10.
Inter-universality:
changes
of
universe
as
changes
of
coordinates
One
fundamental
aspect
of
the
links
[cf.
the
discussion
of
§2.7,
(i)]
—
namely,
the
Θ-link
and
log-link
—
that
occur
in
inter-universal
Teichmüller
theory
is
their
incom-
patibility
with
the
ring
structures
of
the
rings
and
schemes
that
appear
in
their
domains
and
codomains.
In
particular,
when
one
considers
the
result
of
transporting
an
étale-like
structure
such
as
a
Galois
group
[or
étale
fundamental
group]
across
such
28
Shinichi
Mochizuki
a
link
[cf.
the
discussion
of
§2.7,
(iii)],
one
must
abandon
the
interpretation
of
such
a
Galois
group
as
a
group
of
automorphisms
of
some
ring
[or
field]
structure
[cf.
[AbsTopIII],
Remark
3.7.7,
(i);
[IUTchIV],
Remarks
3.6.2,
3.6.3],
i.e.,
one
must
regard
such
a
Galois
group
as
an
abstract
topological
group
that
is
not
equipped
with
any
of
the
“labelling
structures”
that
arise
from
the
relationship
between
the
Galois
group
and
various
scheme-theoretic
objects.
It
is
precisely
this
state
of
affairs
that
results
in
the
quite
central
role
played
in
inter-universal
Teichmüller
theory
by
results
in
[mono-]anabelian
geometry,
i.e.,
by
results
concerned
with
reconstructing
various
scheme-theoretic
structures
from
an
abstract
topological
group
that
“just
happens”
to
arise
from
scheme
theory
as
a
Galois
group/étale
fundamental
group.
In
this
context,
we
remark
that
it
is
also
this
state
of
affairs
that
gave
rise
to
the
term
“inter-universal”:
That
is
to
say,
the
notion
of
a
“universe”,
as
well
as
the
use
of
multiple
universes
within
the
discussion
of
a
single
set-up
in
arithmetic
geometry,
already
occurs
in
the
mathematics
of
the
1960’s,
i.e.,
in
the
mathematics
of
Galois
categories
and
étale
topoi
associated
to
schemes.
On
the
other
hand,
in
this
mathematics
of
the
Grothendieck
school,
typically
one
only
considers
relationships
between
universes
—
i.e.,
between
labelling
apparatuses
for
sets
—
that
are
induced
by
morphisms
of
schemes,
i.e.,
in
essence
by
ring
homomorphisms.
The
most
typical
example
of
this
sort
of
situation
is
the
functor
between
Galois
categories
of
étale
coverings
induced
by
a
morphism
of
connected
schemes.
By
contrast,
the
links
that
occur
in
inter-universal
Teichmüller
theory
are
constructed
by
partially
dismantling
the
ring
structures
of
the
rings
in
their
domains
and
codomains
[cf.
the
discussion
of
§2.7,
(vii)],
hence
necessarily
result
in
much
more
complicated
relationships
between
the
universes
—
i.e.,
be-
tween
the
labelling
apparatuses
for
sets
—
that
are
adopted
in
the
Galois
cat-
egories
that
occur
in
the
domains
and
codomains
of
these
links,
i.e.,
relation-
ships
that
do
not
respect
the
various
labelling
apparatuses
for
sets
that
arise
from
correspondences
between
the
Galois
groups
that
appear
and
the
respective
ring/scheme
theories
that
occur
in
the
domains
and
codomains
of
the
links.
That
is
to
say,
it
is
precisely
this
sort
of
situation
that
is
referred
to
by
the
term
“inter-universal”.
Put
another
way,
a
change
of
universe
may
be
thought
of
[cf.
the
discussion
of
§2.7,
(i)]
as
a
sort
of
abstract/combinatorial/arithmetic
version
of
the
classical
notion
of
a
“change
of
coordinates”.
In
this
context,
it
is
perhaps
of
interest
to
observe
that,
from
a
purely
classical
point
of
view,
the
notion
of
a
[physical]
“universe”
was
typically
visualized
as
a
copy
of
Euclidean
three-space.
Thus,
from
this
classical
point
of
view,
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
29
a
“change
of
universe”
literally
corresponds
to
a
“classical
change
of
the
co-
ordinate
system
—
i.e.,
the
labelling
apparatus
—
applied
to
label
points
in
Euclidean
three-space”!
Indeed,
from
an
even
more
elementary
point
of
view,
perhaps
the
simplest
example
of
the
essential
phenomenon
under
consideration
here
is
the
following
purely
combinatorial
phenomenon:
Consider
the
string
of
symbols
010
—
i.e.,
where
“0”
and
“1”
are
to
be
understood
as
formal
symbols.
Then,
from
the
point
of
view
of
the
length
two
substring
01
on
the
left,
the
digit
“1”
of
this
substring
may
be
specified
by
means
of
its
“coordinate
relative
to
this
substring”,
namely,
as
the
symbol
to
the
far
right
of
the
substring
01.
In
a
similar
vein,
from
the
point
of
view
of
the
length
two
substring
10
on
the
right,
the
digit
“1”
of
this
substring
may
be
specified
by
means
of
its
“coordinate
relative
to
this
substring”,
namely,
as
the
symbol
to
the
far
left
of
the
substring
10.
On
the
other
hand,
neither
of
these
specifications
via
“substring-based
coordinate
systems”
is
meaningful
to
the
opposite
length
two
substring;
that
is
to
say,
only
the
solitary
abstract
symbol
“1”
is
simultaneously
meaningful,
as
a
device
for
specifying
the
digit
of
interest,
relative
to
both
of
the
“substring-based
coordinate
systems”.
Finally,
in
passing,
we
note
that
this
discussion
applies,
albeit
in
perhaps
a
somewhat
∼
trivial
way,
to
the
isomorphism
of
Galois
groups
Ψ
η
X
:
G
K
→
G
K
induced
by
the
Frobenius
morphism
Φ
η
X
in
Example
2.6.1,
(i):
That
is
to
say,
from
the
point
of
view
of
classical
ring
theory,
this
isomorphism
of
Galois
groups
is
easily
seen
to
coincide
with
the
identity
automorphism
of
G
K
.
On
the
other
hand,
if
one
takes
the
point
of
view
that
elements
of
various
subquotients
of
G
K
are
equipped
with
labels
that
arise
from
the
isomorphisms
ρ
or
κ
of
Example
2.6.1,
(ii),
(iii),
i.e.,
from
the
reciprocity
map
of
class
field
theory
or
Kummer
theory,
then
one
must
regard
such
labelling
apparatuses
as
being
incompatible
with
the
Frobenius
morphism
Φ
η
X
.
Thus,
from
this
point
of
view,
the
isomorphism
Φ
η
X
must
be
regarded
as
a
“mysterious,
indeterminate
isomorphism”
[cf.
the
discussion
of
§2.7,
(iii)].
§
2.11.
The
two
underlying
combinatorial
dimensions
of
a
ring
Before
proceeding,
we
pause
to
examine
in
more
detail
the
two
underlying
com-
binatorial
dimensions
of
a
ring
discussed
in
§2.7,
(vii)
[cf.
also
[AbsTopIII],
§I3].
One
way
of
expressing
these
two
underlying
combinatorial
dimensions
of
a
ring
—
i.e.,
constituted
by
addition
and
multiplication
—
is
by
means
of
semi-direct
product
Shinichi
Mochizuki
30
groups
such
as
Z
l
Z
×
l
or
F
l
F
×
l
—
where
l
is
a
prime
number;
Z
l
denotes,
by
abuse
of
notation,
the
underlying
additive
profinite
group
of
the
ring
“Z
l
”
of
l-adic
integers;
Z
×
l
denotes
the
multiplicative
profinite
group
of
invertible
l-adic
integers;
F
l
,
by
abuse
of
notation,
denotes
the
underlying
additive
group
of
the
finite
field
“F
l
”
of
l
elements;
F
×
l
denotes
the
multiplicative
group
×
×
of
the
field
F
l
;
Z
l
,
F
l
act
on
Z
l
,
F
l
via
the
ring
structure
of
Z
l
,
F
l
.
Here,
we
note
that
both
[the
rings]
Z
l
and
F
l
are
closely
related
to
the
fundamental
ring
Z.
Indeed,
Z
may
be
regarded
as
a
dense
subring
of
Z
l
,
while
F
l
may
be
regarded
as
a
“good
finite
discrete
approximation”
of
Z
whenever
l
is
“large”
by
comparison
to
the
numbers
def
of
interest.
Note,
moreover,
that
if
G
k
=
Gal(k/k)
is
the
absolute
Galois
group
of
a
mixed-characteristic
local
field
[i.e.,
“MLF”]
k
of
residue
characteristic
p
for
which
k
is
an
algebraic
closure,
then
the
maximal
tame
quotient
G
k
G
tm
k
is
isomorphic
to
some
open
subgroup
of
the
closed
subgroup
of
the
direct
product
Z
l
Z
×
l
l
=
p
given
by
the
inverse
image
via
the
quotient
Z
l
Z
×
Z
×
l
l
l
=
p
l
=
p
of
the
closed
subgroup
topologically
generated
by
the
image
of
p
[cf.
[NSW],
Proposition
7.5.1].
Thus,
if
we
assume
that
p
=
l,
then
l
may
be
thought
of
as
one
of
the
“l
’s”
in
the
last
two
displays.
In
particular,
from
a
purely
cohomological
point
of
view,
the
two
×
combinatorial
dimensions
“Z
l
”
and
“Z
×
l
”
of
the
semi-direct
product
group
Z
l
Z
l
—
i.e.,
which
correspond
to
the
additive
and
multiplicative
structures
of
the
ring
Z
l
—
may
be
thought
of
as
corresponding
directly
to
the
two
l-cohomological
dimensions
[cf.
[NSW],
Theorem
7.1.8,
(i)]
of
the
profinite
group
G
tm
k
or,
equivalently
[since
l
=
p],
of
the
profinite
group
G
k
.
This
suggests
the
point
of
view
that
the
“restriction
to
l”
should
not
be
regarded
as
essential,
i.e.,
that
one
should
regard
the
two
underlying
combinatorial
dimensions
of
the
ring
k
as
correspond-
ing
to
the
two
cohomological
dimensions
of
its
absolute
Galois
group
G
k
,
and
indeed,
more
generally,
since
the
two
cohomological
dimensions
of
the
absolute
Galois
group
G
F
of
a
[say,
for
simplicity,
totally
imaginary]
number
field
F
[cf.
[NSW],
Proposition
8.3.17]
may
be
thought
of,
via
the
well-known
classical
theory
of
the
Brauer
group,
as
globalizations
[cf.
[NSW],
Corollary
8.1.16;
[NSW],
Theorem
8.1.17]
of
the
two
cohomological
dimensions
of
the
absolute
Galois
groups
of
its
[say,
for
simplicity,
nonarchimedean]
localizations,
that
one
should
regard
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
31
the
two
underlying
combinatorial
dimensions
of
a
[totally
imaginary]
number
field
F
as
corresponding
to
the
two
cohomological
dimensions
of
its
absolute
Galois
group
G
F
.
Moreover,
in
the
case
of
the
local
field
k,
the
two
cohomological
dimensions
of
G
k
may
be
thought
of
as
arising
[cf.,
e.g.,
the
proof
of
[NSW],
Theorem
7.1.8,
(i)]
from
the
one
(
∼
cohomological
dimension
of
the
maximal
unramified
quotient
G
k
G
unr
=
Z)
and
the
k
def
one
cohomological
dimension
of
the
inertia
subgroup
I
k
=
Ker(G
k
G
unr
k
).
Since
I
k
unr
and
G
k
may
be
thought
of
as
corresponding,
via
local
class
field
theory
[cf.
[NSW],
Theorem
7.2.3],
or,
alternatively
[i.e.,
“dually”
—
cf.
[NSW],
Theorem
7.2.6],
via
Kum-
mer
theory,
to
the
subquotients
of
the
multiplicative
group
k
×
associated
to
k
given
by
the
unit
group
O
k
×
and
the
value
group
k
×
/O
k
×
of
k
[together
with
the
corresponding
subquotients
associated
to
the
various
subextensions
of
k
in
k],
we
conclude
that
it
is
natural
to
regard
the
two
underlying
combinatorial
dimensions
of
the
ring
k,
or,
alternatively,
the
two
cohomological
dimensions
of
its
absolute
Galois
group
G
k
,
as
corresponding
to
the
natural
exact
sequence
1
→
O
k
×
→
k
×
→
k
×
/O
k
×
→
1
—
i.e.,
to
the
[non-split,
i.e.,
at
least
when
subject
to
the
requirement
of
func-
toriality
with
respect
to
the
operation
of
passing
to
finite
extensions
of
k!]
“decomposition”
of
k
×
into
its
unit
group
O
k
×
and
value
group
k
×
/O
k
×
.
This
situation
is
reminiscent
of
the
[split!]
decomposition
of
the
multiplicative
topologi-
cal
group
C
×
associated
to
the
field
of
complex
numbers,
i.e.,
which
is
equipped
with
a
natural
decomposition
∼
C
×
→
S
1
×
R
>0
as
a
direct
product
of
its
unit
group
S
1
and
value
group
R
>0
[i.e.,
the
multiplicative
group
of
positive
real
numbers].
§
2.12.
Mono-anabelian
transport
for
mixed-characteristic
local
fields
The
discussion
of
the
two
underlying
combinatorial
dimensions
of
a
ring
in
§2.11
—
especially,
in
the
case
of
an
MLF
“k”—
leads
naturally,
from
the
point
of
view
of
the
analogy
discussed
in
§2.2,
§2.3,
§2.4,
§2.5,
and
§2.7
with
the
classical
theory
of
§1.4
and
§1.5,
to
consideration
of
the
following
examples,
which
may
be
thought
of
as
arithmetic
analogues
of
the
discussion
in
Step
7
of
§1.5
of
the
effect
of
upper
triangular
linear
transformations
and
rotations
on
“local
masses”.
As
one
might
expect
from
the
discussion
of
§2.7,
Kummer
theory
—
i.e.,
applied
to
relate
Frobenius-like
structures
Shinichi
Mochizuki
32
to
their
étale-like
counterparts
—
and
cyclotomic
rigidity
isomorphisms
play
a
central
role
in
these
examples.
In
the
following
examples,
we
use
the
notation
of
§2.11
for
“k”
and
various
objects
related
to
k;
also,
we
shall
write
(O
k
×
⊆)
O
k
(⊆
k
×
)
for
the
topological
multiplicative
monoid
of
nonzero
integral
elements
of
k,
μ
k
⊆
O
k
for
the
topological
module
of
torsion
elements
of
O
k
,
and
∼
ρ
k
:
k
×
→
(k
×
)
∧
→
G
ab
k
for
the
composite
of
the
embedding
of
k
×
into
its
profinite
completion
(k
×
)
∧
with
the
natural
isomorphism
[i.e.,
which
arises
from
local
class
field
theory]
of
(k
×
)
∧
with
the
abelianization
G
ab
k
of
G
k
.
Here,
we
recall,
from
local
class
field
theory,
that
ρ
k
is
functorial
with
respect
to
passage
to
finite
subextensions
of
k
in
k
and
the
Verlagerung
homomorphism
between
abelianizations
of
open
subgroups
of
G
k
.
Also,
we
recall
[cf.,
e.g.,
[AbsAnab],
Proposition
1.2.1,
(iii),
(iv)]
that
the
images
ρ
k
(μ
k
)
⊆
ρ
k
(O
k
)
⊆
×
via
ρ
k
may
be
constructed
group-theoretically
ρ
k
(k
×
)
⊆
G
ab
k
of
μ
k
,
O
k
,
and
k
from
the
topological
group
G
k
.
The
notation
introduced
so
far
for
various
objects
related
to
k
will
also
be
applied
to
finite
subextensions
of
k
in
k,
as
well
as
[i.e.,
by
passing
to
suitable
inductive
limits]
to
k
itself.
In
particular,
if
we
write
μ
k
(G
k
)
⊆
O
k
(G
k
)
⊆
×
k
(G
k
)
for
the
respective
inductive
systems
[or,
by
abuse
of
notation,
when
there
is
no
fear
of
confusion,
inductive
limits],
relative
to
the
Verlagerung
homomorphism
between
abelian-
izations
of
open
subgroups
of
G
k
,
of
the
[group-theoretically
constructible!]
submonoids
ρ
k
(μ
k
)
⊆
ρ
k
(O
k
)
⊆
ρ
k
((k
)
×
)
⊆
G
ab
k
associated
to
the
various
open
subgroups
G
k
⊆
G
k
[i.e.,
where
k
ranges
over
the
finite
subextensions
of
k
in
k],
then
the
various
ρ
k
determine
natural
isomorphisms
∼
ρ
μ
k
:
μ
k
→
μ
k
(G
k
),
∼
ρ
O
:
O
k
→
O
k
(G
k
),
k
ρ
k
×
:
k
×
∼
×
→
k
(G
k
)
def
of
[multiplicative]
G
k
-monoids.
In
the
following,
we
shall
also
use
the
notation
μ
k
Z
=
def
Hom(Q/Z,
μ
k
)
and
μ
k
Z
(G
k
)
=
Hom(Q/Z,
μ
k
(G
k
)).
Example
2.12.1.
Nonarchimedean
multiplicative
monoids
of
local
in-
tegers.
(i)
In
the
following,
we
wish
to
regard
the
pair
“G
k
O
k
”
as
an
abstract
ind-
topological
monoid
“O
k
”
[i.e.,
inductive
system
of
topological
monoids]
equipped
with
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
33
a
continuous
action
by
an
abstract
topological
group
“G
k
”.
Thus,
for
instance,
we
may
×
think
of
k
as
the
groupification
(O
k
)
gp
of
the
monoid
O
k
,
of
μ
k
as
the
subgroup
of
torsion
elements
of
the
monoid
O
k
,
and
of
O
k
⊆
k
×
as
the
result
of
considering
the
G
k
-
×
×
invariants
(O
k
)
G
k
⊆
(k
)
G
k
of
the
inclusion
O
k
⊆
k
.
Observe
that,
by
considering
the
action
of
G
k
on
the
various
N
-th
roots,
for
N
a
positive
integer,
of
elements
of
k
×
,
we
obtain
a
natural
Kummer
map
κ
k
:
k
×
→
H
1
(G
k
,
μ
k
Z
)
—
which
may
be
composed
with
the
natural
isomorphism
ρ
μ
k
to
obtain
a
natural
em-
bedding
:
k
×
→
H
1
(G
k
,
μ
k
Z
(G
k
))
κ
Gal
k
—
where
we
note
that
the
cohomology
module
in
the
codomain
of
this
embedding
may
be
constructed
group-theoretically
from
the
abstract
topological
group
“G
k
”.
On
the
other
hand,
it
follows
immediately
from
the
definitions
that
κ
k
may
be
con-
structed
functorially
from
the
abstract
ind-topological
monoid
with
continuous
topo-
logical
group
action
“G
k
O
k
”.
(ii)
In
fact,
ρ
μ
k
may
also
be
constructed
functorially
from
the
abstract
ind-topological
monoid
with
continuous
topological
group
action
“G
k
O
k
”.
Indeed,
this
follows
formally
from
the
fact
that
∼
there
exists
a
canonical
isomorphism
Q/Z
→
H
2
(G
k
,
μ
k
)
that
may
be
constructed
functorially
from
this
data
“G
k
O
k
”.
—
i.e.,
by
applying
this
functorial
construction
to
both
the
data
“G
k
O
k
”
and
the
data
“G
k
O
k
(G
k
)”
and
then
observing
that
ρ
μ
k
may
be
characterized
as
the
unique
∼
∼
isomorphism
μ
k
→
μ
k
(G
k
)
that
is
compatible
with
the
isomorphisms
Q/Z
→
H
2
(G
k
,
μ
k
)
∼
and
Q/Z
→
H
2
(G
k
,
μ
k
(G
k
)).
To
construct
this
canonical
isomorphism
[cf.,
e.g.,
the
proof
of
[AbsAnab],
Proposition
1.2.1,
(vii);
the
statement
and
proof
of
[FrdII],
The-
orem
2.4,
(ii);
the
statement
of
[AbsTopIII],
Corollary
1.10,
(i),
(a);
the
statement
of
×
[AbsTopIII],
Proposition
3.2,
(i),
for
more
details],
we
first
observe
that
since
k
/μ
k
is
a
×
∼
Q-vector
space,
it
follows
that
we
have
natural
isomorphisms
H
2
(G
k
,
μ
k
)
→
H
2
(G
k
,
k
),
×
∼
H
2
(I
k
,
μ
k
)
→
H
2
(I
k
,
k
).
Since,
moreover,
the
inertia
subgroup
I
k
⊆
G
k
is
of
co-
×
homological
dimension
1
[cf.
the
discussion
of
§2.11],
we
conclude
that
H
2
(I
k
,
k
)
∼
→
H
2
(I
k
,
μ
k
)
=
0.
Next,
let
us
recall
that,
by
elementary
Galois
theory
[i.e.,
“Hilbert’s
×
Theorem
90”],
one
knows
that
H
1
(I
k
,
k
)
=
0.
Thus,
we
conclude
from
the
Leray-
→
1
that,
Serre
spectral
sequence
associated
to
the
extension
1
→
I
k
→
G
k
→
G
unr
k
Shinichi
Mochizuki
34
if
we
write
k
unr
⊆
k
for
the
subfield
of
I
k
-invariants
of
k,
then
we
have
a
natural
iso-
×
∼
unr
×
)
).
On
the
other
hand,
the
valuation
map
morphism
H
2
(G
k
,
k
)
→
H
2
(G
unr
k
,
(k
unr
×
unr
×
∼
)
)
→
H
2
(G
unr
on
(k
)
determines
an
isomorphism
H
2
(G
unr
k
,
(k
k
,
Z)
[where
again
we
apply
“Hilbert’s
Theorem
90”,
this
time
to
the
residue
field
of
k].
Moreover,
by
∼
→
Z
[which
is
applying
the
isomorphism
determined
by
the
Frobenius
element
G
unr
k
group-theoretically
constructible
—
cf.
[AbsAnab],
Proposition
1.2.1,
(iv)],
together
with
the
long
exact
sequence
in
Galois
cohomology
associated
to
the
short
exact
se-
quence
0
→
Z
→
Q
→
Q/Z
→
0
[and
the
fact
that
Q
is
a
Q-vector
space!],
we
obtain
∼
∼
1
unr
a
natural
isomorphism
H
2
(G
unr
k
,
Z)
→
H
(G
k
,
Q/Z)
→
Hom(
Z,
Q/Z)
=
Q/Z.
Thus,
by
taking
the
inverse
of
the
composite
of
the
various
natural
isomorphisms
constructed
so
far
[solely
from
the
data
“G
k
O
k
”!],
we
obtain
the
desired
canonical
isomorphism
∼
Q/Z
→
H
2
(G
k
,
μ
k
).
(iii)
The
functorial
construction
given
in
(ii)
—
i.e.,
via
well-known
elementary
techniques
involving
Brauer
groups
of
the
sort
that
appear
in
local
class
field
theory
—
of
the
cyclotomic
rigidity
isomorphism
ρ
μ
k
from
the
data
“G
k
O
k
”
is
perhaps
the
most
fundamental
case
—
at
least
in
the
context
of
the
arithmetic
of
NF’s
and
MLF’s
—
of
the
phenomenon
of
cyclotomic
rigidity.
One
formal
consequence
of
the
discussion
of
(i),
(ii)
is
the
fact
that
the
operation
of
passing
from
the
data
“G
k
O
k
”
to
the
data
“G
k
”
is
fully
faithful,
i.e.,
one
has
a
natural
bijection
Aut(G
k
O
k
)
∼
→
Aut(G
k
)
—
where
the
first
“Aut(−)”
denotes
automorphisms
of
the
data
“G
k
O
k
”
consist-
ing
of
an
abstract
ind-topological
monoid
with
continuous
topological
group
action;
the
second
“Aut(−)”
denotes
automorphisms
of
the
data
“G
k
”
consisting
of
an
abstract
topological
group
[cf.
[AbsTopIII],
Proposition
3.2,
(iv)].
Indeed,
surjectivity
follows
for-
mally
from
the
functorial
construction
of
the
data
“G
k
O
k
(G
k
)”
from
the
abstract
topological
group
G
k
[cf.
the
discussion
at
the
beginning
of
the
present
§2.12];
injectivity
follows
formally
from
the
fact
that,
as
a
consequence
of
the
cyclotomic
rigidity
discussed
in
(ii),
one
has
a
functorial
construction
from
the
data
“G
k
O
k
”
of
the
embedding
κ
Gal
:
k
×
→
H
1
(G
k
,
μ
k
Z
(G
k
))
[in
fact
applied
in
the
case
where
“k”
is
replaced
by
arbi-
k
trary
finite
subextensions
of
k
in
k]
into
the
container
H
1
(G
k
,
μ
k
Z
(G
k
))
[which
may
be
constructed
solely
from
the
abstract
topological
group
G
k
!].
Note
that
this
situation
may
also
be
understood
in
terms
of
the
general
framework
of
mono-anabelian
transport
discussed
in
§2.7,
(v)
[cf.
also
Example
2.6.1,
(iii),
(iv)],
by
considering
the
commutative
diagram
∼
H
1
(G
k
,
μ
k
Z
(G
k
))
−→
H
1
(G
k
,
μ
k
Z
(G
k
))
⏐
κ
Gal
|
⏐
k
O
k
O
k
−→
∼
⏐
κ
Gal
|
⏐
k
O
k
O
k
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
35
—
where
the
horizontal
arrows
are
induced
by
some
given
automorphism
of
the
data
“G
k
O
k
”;
the
vertical
arrows
serve
to
embed
the
Frobenius-like
data
“O
k
”
into
the
étale-like
container
H
1
(G
k
,
μ
k
Z
(G
k
)).
Finally,
we
observe
that
the
cyclotomic
rigidity
discussed
in
(ii)
may
be
understood,
relative
to
the
exact
sequence
1
→
O
k
×
→
k
×
×
→
k
/O
k
×
(
∼
=
Q)
→
1
—
which,
as
was
discussed
in
the
final
portion
of
§2.11,
may
be
thought
of
as
corre-
sponding
to
the
two
underlying
combinatorial
dimensions
of
the
ring
k
—
as
revolving
×
×
×
around
the
rigidity
of
the
two
fundamental
subquotients
μ
k
⊆
k
and
k
k
/O
k
×
×
of
k
.
When
viewed
in
this
light,
the
discussion
of
the
present
Example
2.12.1
may
be
thought
of,
relative
to
the
analogy
discussed
in
§2.2,
§2.3,
§2.4,
§2.5,
and
§2.7
with
the
classical
theory
of
§1.4
and
§1.5,
as
corresponding
to
the
discussion
of
the
effect
on
“lo-
cal
masses”
of
the
unipotent
linear
transformations
that
appeared
in
the
discussion
of
Step
7
of
§1.5.
(iv)
At
this
point,
it
is
perhaps
of
interest
to
observe
that
there
is
an
alternative
ap-
proach
to
constructing
the
cyclotomic
rigidity
isomorphism
ρ
μ
k
.
That
is
to
say,
instead
of
reasoning
as
in
(ii),
one
may
reason
as
follows.
First,
we
observe
that,
by
applying
the
functorial
construction
of
(ii)
in
the
case
of
the
data
“G
k
O
k
(G
k
)”,
one
obtains
∼
a
canonical
isomorphism
Q/Z
→
H
2
(G
k
,
μ
k
Z
(G
k
)).
Since
the
cup
product
in
group
co-
homology,
together
with
this
canonical
isomorphism,
determines
a
perfect
duality
[cf.
∼
Z
1
[NSW],
Theorem
7.2.6],
one
thus
obtains
a
natural
isomorphism
G
ab
k
→
H
(G
k
,
μ
k
(G
k
)).
Write
O
k
(G
k
)
Kum
⊆
H
1
(G
k
,
μ
k
Z
(G
k
))
for
the
image
via
this
natural
isomorphism
of
O
k
(G
k
)
⊆
G
ab
k
[i.e.,
the
submodule
of
Kum
may
be
constructed
group-theoretically
G
k
-invariants
of
O
k
(G
k
)].
Thus,
O
k
(G
k
)
from
the
abstract
topological
group
G
k
.
Next,
let
us
recall
the
elementary
fact
that,
relative
to
the
natural
inclusion
Q
→
Z
⊗
Q,
we
have
an
equality
Z
×
=
{1}
Q
>0
—
where
Q
>0
⊆
Q
denotes
the
multiplicative
monoid
of
positive
rational
numbers.
Now
let
us
observe
that
it
follows
formally
from
this
elementary
fact
—
for
instance,
by
considering
the
quotient
O
k
(G
k
)
O
k
(G
k
)/O
k
×
(G
k
)
(
∼
=
N)
by
the
submonoid
of
units
×
×
O
k
(G
k
)
⊆
O
k
(G
k
)
—
that
the
only
element
∈
Z
that,
relative
to
the
natural
action
of
Z
×
on
H
1
(G
k
,
μ
k
Z
(G
k
))
[i.e.,
induced
by
the
natural
action
of
Z
×
on
μ
k
(G
k
)],
preserves
the
submonoid
O
k
(G
k
)
is
the
identity
element
1
∈
Z
×
.
In
particular,
it
follows
that
the
cyclotomic
rigidity
isomorphism
ρ
μ
k
may
be
characterized
as
the
unique
∼
isomorphism
μ
k
→
μ
k
(G
k
)
that
is
compatible
with
the
submonoids
κ
k
(O
k
)
⊆
H
1
(G
k
,
μ
k
Z
)
and
O
k
(G
k
)
Kum
⊆
H
1
(G
k
,
μ
k
Z
(G
k
)).
36
Shinichi
Mochizuki
This
characterization
thus
yields
an
alternative
approach
to
the
characterization
of
the
cyclotomic
rigidity
isomorphism
ρ
μ
k
given
in
(ii)
[cf.
the
discussion
of
[IUTchIII],
Re-
mark
2.3.3,
(viii)].
On
the
other
hand,
there
is
a
fundamental
difference
between
this
alternative
approach
and
the
approach
of
(ii):
Indeed,
one
verifies
immediately
that
the
approach
of
(ii)
is
compatible
with
the
profinite
topology
of
G
k
in
the
sense
that
the
construction
of
(ii)
may
be
formulated
as
the
result
of
applying
a
suitable
limit
operation
to
“finite
versions”
of
this
construction
of
(ii),
i.e.,
versions
in
which
“G
k
”
is
replaced
by
the
quotients
of
“G
k
”
by
sufficiently
small
normal
open
subgroups
of
“G
k
”,
and
“O
k
”
is
replaced
by
the
submonoids
of
invariants
with
respect
to
such
normal
open
subgroups.
By
contrast,
the
alternative
approach
just
discussed
is
fundamen-
tally
incompatible
with
the
profinite
topology
of
G
k
in
the
sense
that
the
crucial
fact
×
Z
=
{1}
—
which
may
be
thought
of
as
a
sort
of
discreteness
property
[cf.
Q
>0
the
discussion
of
[IUTchIII],
Remark
3.12.1,
(iii);
[IUTchIV],
Remark
2.3.3,
(ii)]
—
may
only
be
applied
at
the
level
of
the
full
profinite
group
G
k
[i.e.,
at
the
level
of
Kummer
classes
with
coefficients
in
some
copy
of
Z(1)],
not
at
the
level
of
finite
quotients
of
G
k
[i.e.,
at
the
level
of
Kummer
classes
with
coefficients
in
some
finite
quotient
of
some
copy
of
Z(1)].
Thus,
in
summary,
although
this
alternative
approach
has
the
disadvantage
of
being
incompatible
with
the
profinite
topology
of
G
k
,
various
versions
of
this
approach
—
i.e.,
involving
constructions
that
depend,
in
an
essential
way,
on
the
crucial
×
Z
=
{1}
—
will,
nevertheless,
play
an
important
role
in
inter-universal
fact
Q
>0
Teichmüller
theory
[cf.
the
discussion
of
Example
2.13.1
below].
Example
2.12.2.
Frobenius
morphisms
on
nonarchimedean
multiplica-
tive
monoids
of
local
integers.
(i)
One
way
to
gain
a
further
appreciation
of
the
cyclotomic
rigidity
phenomenon
discussed
in
Example
2.12.1
is
to
consider
the
pair
“G
k
O
k
×
”,
which
again
we
re-
gard
as
consisting
of
an
abstract
ind-topological
monoid
“O
k
×
”
[i.e.,
inductive
system
of
topological
monoids]
equipped
with
a
continuous
action
by
an
abstract
topological
group
“G
k
”.
Since
O
k
×
may
be
thought
of
as
an
inductive
system/limit
of
profinite
abelian
groups,
it
follows
immediately
that
there
is
a
natural
G
k
-equivariant
action
of
Z
×
on
the
data
“G
k
O
k
×
”.
Moreover,
if
α
is
an
arbitrary
automorphism
of
this
data
“G
k
O
k
×
”
[i.e.,
regarded
as
an
abstract
ind-topological
monoid
equipped
with
a
continuous
action
by
an
abstract
topological
group],
then
although
it
is
not
necessarily
the
case
that
α
is
∼
compatible
with
the
cyclotomic
rigidity
isomorphism
ρ
μ
k
:
μ
k
→
μ
k
(G
k
),
one
verifies
immediately
[from
the
fact
that,
as
an
abstract
abelian
group,
μ
k
∼
=
Q/Z,
together
with
×
the
elementary
fact
that
Aut(Q/Z)
=
Z
]
that
there
always
exists
a
unique
element
λ
∈
Z
×
such
that
the
automorphism
λ
·
α
of
the
data
“G
k
O
k
×
”
is
compatible
with
ρ
μ
k
.
Thus,
by
arguing
as
in
Example
2.12.1,
(iii),
one
concludes
that
one
has
a
natural
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
bijection
Aut(G
k
O
k
×
)
∼
→
37
Z
×
×
Aut(G
k
)
—
where
the
first
“Aut(−)”
denotes
automorphisms
of
the
data
“G
k
O
k
×
”
consist-
ing
of
an
abstract
ind-topological
monoid
with
continuous
topological
group
action;
the
second
“Aut(−)”
denotes
automorphisms
of
the
data
“G
k
”
consisting
of
an
abstract
topological
group
[cf.
[AbsTopIII],
Proposition
3.3,
(ii);
[FrdII],
Remark
2.4.2].
Just
as
in
the
case
of
Example
2.12.1,
(iii),
this
situation
may
also
be
understood
in
terms
of
the
general
framework
of
mono-anabelian
transport
discussed
in
§2.7,
(v)
[cf.
also
Example
2.6.1,
(iii),
(iv)],
by
considering
the
commutative
diagram
∼
H
1
(G
k
,
μ
k
Z
(G
k
))
−→
H
1
(G
k
,
μ
k
Z
(G
k
))
⏐
κ
Gal
|
×
⏐
k
O
k
?
O
k
×
−→
∼
⏐
κ
Gal
|
×
⏐
k
O
k
O
k
×
—
where
the
horizontal
arrows
are
induced
by
some
given
automorphism
of
the
data
“G
k
O
k
×
”;
the
vertical
arrows
serve
to
embed
the
Frobenius-like
data
“O
k
×
”
into
the
étale-like
container
H
1
(G
k
,
μ
k
Z
(G
k
));
the
diagram
commutes
[cf.
“
?”]
up
to
the
action
of
a
suitable
element
∈
Z
×
.
(ii)
Let
π
k
∈
O
k
be
a
uniformizer
of
O
k
.
Then
one
sort
of
intermediate
type
of
data
between
the
data
“G
k
O
k
×
”
considered
in
(i)
above
and
the
data
“G
k
O
k
”
con-
sidered
in
Example
2.12.1
is
the
data
“G
k
O
k
×
·
O
k
(⊆
O
k
)”,
which
again
we
regard
as
consisting
of
an
abstract
ind-topological
monoid
“O
k
×
·
O
k
”
[i.e.,
inductive
system
of
topological
monoids]
equipped
with
a
continuous
action
by
an
abstract
topological
group
“G
k
”.
Here,
we
observe
that
O
k
×
·
O
k
=
O
k
×
·
π
k
N
.
Let
Z
N
≥
2,
α
∈
Aut(G
k
O
k
×
).
Then
observe
that
N
,
α
determine
—
i.e.,
in
the
spirit
of
the
discussion
of
§2.4
—
a
sort
of
Frobenius
morphism
φ
N,α
G
k
O
k
×
·
O
k
→
G
k
O
k
×
·
O
k
π
k
→
π
k
N
that
restricts
to
α
on
the
data
“G
k
O
k
×
”.
From
the
point
of
view
of
the
general
framework
of
mono-anabelian
transport
discussed
in
§2.7,
(v)
[cf.
also
Example
2.6.1,
(iii),
(iv)],
this
sort
of
Frobenius
morphism
φ
N,α
induces
a
commutative
diagram
∼
H
1
(G
k
,
μ
k
Z
(G
k
))
−→
H
1
(G
k
,
μ
k
Z
(G
k
))
⏐
κ
Gal
|
⏐
k
O
k
?
⏐
κ
Gal
|
⏐
k
O
k
O
k
−→
O
k
38
Shinichi
Mochizuki
—
where
the
horizontal
arrows
are
induced
by
φ
N,α
;
the
vertical
arrows
serve
to
embed
the
Frobenius-like
data
“O
k
”
into
the
étale-like
container
H
1
(G
k
,
μ
k
Z
(G
k
));
the
diagram
commutes
[cf.
“
?”]
up
to
the
action
of
a
suitable
element
∈
Z
×
on
O
k
×
⊆
O
k
and
a
suitable
element
∈
N
[namely,
N
∈
N]
on
π
k
N
.
Finally,
we
observe
that
the
diagonal
nature
of
the
action
of
φ
N,α
on
the
unit
group
O
k
×
[via
α]
and
the
value
group
π
k
Z
[by
raising
to
the
N
-th
power]
portions
of
the
ind-topological
monoid
O
k
×
·O
k
may
be
thought
of,
relative
to
the
analogy
discussed
in
§2.2,
§2.3,
§2.4,
§2.5,
and
§2.7
with
the
classical
theory
of
§1.4
and
§1.5,
as
corresponding
to
the
discussion
of
the
effect
on
“local
masses”
of
the
toral
dilations
that
appeared
in
the
discussion
of
Step
7
of
§1.5.
Example
2.12.3.
Nonarchimedean
logarithms.
(i)
The
discussion
of
various
simple
cases
of
mono-anabelian
transport
in
Examples
2.12.1,
(iii);
2.12.2,
(i),
(ii),
concentrated
on
the
Kummer-theoretic
aspects,
i.e.,
in
effect,
on
the
Kummer-detachment
indeterminacies
[cf.
§2.7,
(vi)],
or
lack
thereof,
of
the
examples
considered.
On
the
other
hand,
another
fundamental
aspect
of
these
examples
[cf.
the
natural
bijections
of
Examples
2.12.1,
(iii);
2.12.2,
(i)]
is
the
étale-transport
indeterminacies
[cf.
§2.7,
(vi)]
that
occur
as
a
result
of
the
well-known
existence
of
elements
∈
Aut(G
k
)
that
do
not
preserve
the
ring
structure
on
[the
union
with
{0}
of]
O
k
(G
k
)
—
cf.
[NSW],
the
Closing
Remark
preceding
Theorem
12.2.7.
By
contrast,
if
X
is
a
hyperbolic
curve
of
strictly
Belyi
type
[cf.
[AbsTopII],
Definition
3.5]
over
k,
and
we
write
Π
X
for
the
étale
fundamental
group
of
X
[for
a
suitable
choice
of
basepoint],
then
it
follows
from
the
theory
of
[AbsTopIII],
§1
[cf.
[AbsTopIII],
Theorem
1.9;
[AbsTopIII],
Remark
1.9.2;
[AbsTopIII],
Corollary
1.10],
that
if
one
regards
G
k
as
a
quotient
Π
X
G
k
of
Π
X
,
then
there
exists
a
functorial
algorithm
for
reconstructing
this
quotient
Π
X
G
k
of
Π
X
,
together
with
the
ring
structure
on
[the
union
with
{0}
of]
O
k
(G
k
),
from
the
abstract
topological
group
Π
X
.
Here,
we
recall
that
[it
follows
immediately
from
the
definitions
that]
any
connected
finite
étale
covering
of
a
once-punctured
elliptic
curve
[i.e.,
an
elliptic
curve
minus
the
origin]
over
k
that
is
defined
over
an
NF
is
necessarily
of
strictly
Belyi
type.
(ii)
Write
(O
k
×
)
pf
for
the
perfection
[cf.,
e.g.,
[FrdI],
§0]
of
the
ind-topological
monoid
O
k
×
.
Thus,
it
follows
immediately
from
the
elementary
theory
of
p-adic
fields
[cf.,
e.g.,
[Kobl],
Chapter
IV,
§1,
§2]
that
the
p-adic
logarithm
determines
a
G
k
-equivariant
bijection
∼
log
k
:
(O
k
×
)
pf
→
k
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
39
with
respect
to
which
the
operation
of
multiplication,
which
we
shall
often
denote
by
the
notation
“”,
in
the
domain
corresponds
to
the
operation
of
addition,
which
we
shall
often
denote
by
the
notation
“”,
in
the
codomain.
This
bijection
fits
into
a
diagram
...
O
k
⊇
O
k
×
∼
(O
k
×
)
pf
→
k
⊇
O
k
...
—
where
the
“.
.
.”
on
the
left
and
right
denote
the
result
of
juxtaposing
copies
of
the
portion
of
the
diagram
“from
O
k
to
O
k
”,
i.e.,
copies
that
are
glued
together
along
the
initial/final
instances
of
“O
k
”.
Here,
we
observe
that
the
various
objects
that
appear
in
this
diagram
may
be
regarded
as
being
equipped
with
a
natural
action
of
Π
X
[for
X
as
in
(i)],
which
acts
via
the
natural
quotient
Π
X
G
k
.
To
keep
the
notation
simple,
we
shall
denote
the
portion
of
the
diagram
“from
O
k
to
O
k
”
by
means
of
the
notation
log
:
O
k
→
O
k
.
Thus,
the
diagram
of
the
above
display
may
be
written
∼
...
→
log
.
.
.
−→
Π
X
O
k
∼
→
log
−→
Π
X
O
k
∼
→
log
−→
Π
X
O
k
∼
→
...
log
−→
.
.
.
—
i.e.,
regarded
as
a
sequence
of
iterates
of
“log”.
Here,
since
the
operation
“log”
[i.e.,
which,
in
effect,
converts
“”
into
“”]
is
incompatible
with
the
ring
structures
on
[the
union
with
{0}
of]
the
copies
of
O
k
in
the
domain
and
codomain
of
“log”,
we
observe
—
in
accordance
with
the
discussion
of
§2.10!
—
that
it
is
natural
to
regard
the
∼
various
copies
of
O
k
as
being
equipped
with
distinct
labels
and
the
isomorphisms
“
→
”
between
different
copies
of
Π
X
as
being
indeterminate
isomorphisms
between
dis-
tinct
abstract
topological
groups.
Such
diagrams
are
studied
in
detail
in
[AbsTopIII],
and,
moreover,
form
the
fundamental
model
for
the
log-link
of
inter-universal
Teichmüller
theory
[cf.
§3.3,
(ii),
(vi),
below],
which
is
studied
in
detail
in
[IUTchIII].
(iii)
It
follows
from
the
mono-anabelian
theory
of
[AbsTopIII],
§1
[cf.
[AbsTopIII],
Theorem
1.9;
[AbsTopIII],
Corollary
1.10],
that,
if
we
regard
G
k
as
a
quotient
of
Π
X
,
[cf.
then
the
image,
which
we
denote
by
O
k
(Π
X
)
(⊆
H
1
(G
k
,
μ
k
Z
(G
k
))),
of
O
k
via
κ
Gal
k
Example
2.12.1,
(iii)]
may
be
reconstructed
—
i.e.,
as
a
topological
monoid
equipped
with
a
ring
structure
[on
its
union
with
{0}]
—
from
the
abstract
topological
group
Π
X
.
By
applying
this
construction
to
arbitrary
open
subgroups
of
G
k
and
passing
to
inductive
systems/limits,
we
thus
obtain
an
ind-topological
monoid
O
k
(Π
X
)
equipped
with
a
natural
continuous
action
by
Π
X
and
a
ring
structure
[on
its
union
with
{0}].
Thus,
from
the
point
of
view
of
the
general
framework
of
mono-anabelian
transport
Shinichi
Mochizuki
40
discussed
in
§2.7,
(v)
[cf.
also
Example
2.6.1,
(iii),
(iv)],
we
obtain
a
diagram
∼
O
k
(Π
X
)
−→
O
k
(Π
X
)
...
⏐
⏐
Kum
?
O
k
−→
log
⏐
⏐
Kum
...
O
k
—
where
the
upper
horizontal
arrow
is
induced
by
some
indeterminate
isomorphism
∼
Π
X
→
Π
X
[cf.
the
discussion
of
§2.10];
the
lower
horizontal
arrow
is
the
operation
“log”
discussed
in
(ii);
the
vertical
arrows
are
the
“Kummer
isomorphisms”
determined
by
the
various
“κ
Gal
k
”
associated
to
open
subgroups
of
G
k
;
the
“.
.
.”
denote
iterates
of
the
square
surrounding
the
“
?”.
Thus,
the
vertical
arrows
of
this
diagram
relate
the
various
copies
of
Frobenius-like
data
O
k
to
the
various
copies
of
étale-like
data
O
k
(Π
X
),
which
are
coric
[cf.
the
discussion
of
§2.7,
(iv)]
with
respect
to
the
“link”
constituted
by
the
operation
log.
(iv)
The
diagram
of
(iii)
is,
of
course,
far
from
being
commutative
[cf.
the
notation
“
?”],
i.e.,
at
least
at
the
level
of
images
of
elements
via
the
various
composites
of
arrows
in
the
diagram.
On
the
other
hand,
if,
instead
of
considering
such
images
of
elements
via
composites
of
arrows
in
the
diagram,
one
considers
regions
[i.e.,
subsets]
of
[the
union
with
{0}
of
the
groupification
of]
O
k
(⊆
O
k
)
or
O
k
(Π
X
)
(⊆
O
k
(Π
X
)),
then
one
verifies
easily
that
the
following
observation
holds:
Write
I
=
(2p)
−1
·
log
k
(O
k
×
)
⊆
k
=
{0}
∪
(O
k
)
gp
def
and
I(Π
X
)
⊆
{0}
∪
O
k
(Π
X
)
gp
for
the
corresponding
subset
of
the
union
with
{0}
of
the
groupification
of
O
k
(Π
X
).
Then
we
have
inclusions
of
“regions”
O
k
⊆
I
⊇
log
k
(O
k
×
)
within
I,
as
well
as
corresponding
inclusions
for
I(Π
X
).
The
compact
“region”
I,
which
is
referred
to
as
the
log-shell,
plays
an
important
role
in
inter-universal
Teichmüller
theory.
Note
that
one
has
both
Frobenius-like
[i.e.,
I]
and
étale-like
[i.e.,
I(Π
X
)]
versions
of
the
log-shell.
Here,
we
observe
that,
from
the
point
of
view
of
the
discussion
of
arithmetic
holomorphic
structures
in
§2.7,
(vii),
both
of
these
versions
are
holomorphic
in
the
sense
that
they
depend,
at
least
in
an
a
priori
sense,
on
“log
k
”,
i.e.,
which
is
defined
in
terms
of
a
power
series
that
only
makes
sense
if
one
is
equipped
with
both
“”
and
“”
[i.e.,
both
the
additive
and
multiplicative
structures
of
the
ring
k].
On
the
other
hand,
if
one
writes
“O
×μ
”
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
41
for
the
quotient
of
“O
×
”
by
its
torsion
subgroup
[i.e.,
by
the
roots
of
unity],
then
log
k
determines
natural
bijections
of
topological
modules
∼
O
k
×μ
⊗
Q
→
I
⊗
Q,
∼
O
k
×μ
(Π
X
)
⊗
Q
→
I(Π
X
)
⊗
Q
—
i.e.,
within
which
the
lattices
O
k
×μ
⊗
(2p)
−1
,
O
k
×μ
(Π
X
)
⊗
(2p)
−1
correspond,
respec-
tively,
to
I,
I(Π
X
).
In
particular,
by
applying
these
bijections,
we
may
think
of
the
topological
modules
I,
I(Π
X
)
as
objects
constructed
from
the
topological
modules
O
k
×μ
,
O
k
×μ
(G
k
)
[cf.
the
notation
introduced
at
the
beginning
of
the
present
§2.12;
the
notation
of
Example
2.12.1,
(iv)],
i.e.,
as
objects
constructed
from
mono-analytic
structures
[cf.
the
discussion
of
§2.7,
(vii)].
That
is
to
say,
in
addition
to
the
holomorphic
Frobenius-like
and
holomorphic
étale-like
versions
of
the
log-shell
discussed
above,
one
may
also
consider
mono-
analytic
Frobenius-like
and
mono-analytic
étale-like
versions
of
the
log-shell.
All
four
of
these
versions
of
the
log-shell
play
an
important
role
in
inter-universal
Te-
ichmüller
theory
[cf.
the
discussion
of
§3.6,
(iv),
below;
[IUTchIII],
Definition
1.1,
(i),
(iv);
[IUTchIII],
Proposition
1.2,
(v),
(vi),
(viii),
(ix),
(x);
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob7);
[IUTchIII],
Remark
3.12.2,
(iv),
(v)].
Returning
to
the
issue
of
the
non-
commutativity
[i.e.,
“
?”]
of
the
diagram
of
(iii),
we
observe
the
following:
the
inclusions
of
“regions”
discussed
above
may
be
interpreted
as
asserting
that
the
holomorphic
étale-like
log
shell
I(Π
X
)
serves
as
a
container
for
[i.e.,
as
a
“region”
that
contains]
the
images
—
i.e.,
of
O
k
,
O
k
×
⊆
O
k
,
or,
in
the
case
of
multiple
iterates
of
log,
even
smaller
subsets
of
O
k
×
—
via
all
possible
composites
of
arrows
of
the
diagram
of
(iii)
[including
the
“.
.
.”
on
the
left-
and
right-hand
sides
of
the
diagram!].
This
property
of
the
log-shell
will
be
referred
to
as
upper
semi-commutativity
[cf.
[IUTchIII],
Remark
1.2.2,
(i),
(iii)].
Thus,
this
property
of
upper
semi-commutativity
constitutes
a
sort
of
Kummer-detachment
indeterminacy
[cf.
the
discussion
of
§2.7,
(vi)]
and
may
be
regarded
as
an
answer
to
the
question
of
computing
the
discrepancy
between
the
two
Kummer
theories
in
the
domain
and
codomain
of
the
link
“log”
[cf.
the
discussion
at
the
beginning
of
§2.9].
Another
important
answer,
in
the
context
of
inter-universal
Teichmüller
theory,
to
this
computational
question
is
given
by
the
theory
of
log-volumes
[i.e.,
where
we
use
the
term
log-volume
to
refer
to
the
natural
logarithm
of
the
volume
∈
R
>0
of
a
region]:
There
is
a
natural
definition
of
the
notion
of
the
log-volume
∈
R
of
a
region
[i.e.,
compact
open
subset]
of
k
=
{0}
∪
(O
k
)
gp
,
which
is
normalized
so
that
the
Shinichi
Mochizuki
42
log-volume
of
O
k
is
0,
while
the
log-volume
of
p·O
k
is
−
log(p).
This
log-volume
is
compatible
[in
the
evident
sense]
with
passage
between
the
four
versions
of
log-shells
discussed
above,
as
well
as
with
log
in
the
sense
that
it
assumes
the
same
value
∈
R
on
regions
that
are
mapped
bijectively
to
one
another
via
log
k
[cf.
[AbsTopIII],
Proposition
5.7,
(i);
[IUTchIII],
Proposition
1.2,
(iii);
[IUTchIII],
Proposition
3.9,
(i),
(ii),
(iv);
[IUTchIII],
Remark
3.9.4].
These
properties
of
upper
semi-commutativity
and
log-volume
compatibility
will
be
sufficient
for
the
purposes
of
inter-universal
Teichmüller
theory.
(v)
Finally,
we
observe
that
since
the
operation
log
—
which
maps
and
relates
unit
groups
[cf.
(O
k
×
)
pf
]
to
value
groups
[i.e.,
nonzero
non-units
of
k]
—
may
be
thought
of
as
an
operation
that
“juggles”,
or
“rotates”,
the
two
underlying
combinatorial
dimensions
[cf.
the
discussion
of
§2.11]
of
the
ring
k
[cf.
[AbsTopIII],
§I3],
one
may
think
of
this
operation
log,
relative
to
the
analogy
discussed
in
§2.2,
§2.3,
§2.4,
§2.5,
and
§2.7
with
the
classical
theory
of
§1.4
and
§1.5,
as
corresponding
to
the
discussion
of
the
effect
on
“local
masses”
of
the
rotations
that
appeared
in
the
discussion
of
Step
7
of
§1.5.
§
2.13.
Mono-anabelian
transport
for
monoids
of
rational
functions
Let
k
be
either
an
MLF
or
an
NF;
X
a
hyperbolic
curve
of
strictly
Belyi
type
[cf.
[AbsTopII],
Definition
3.5]
over
k;
K
X
an
algebraic
closure
of
the
function
field
K
X
of
X;
k
⊆
K
X
the
algebraic
closure
of
k
determined
by
K
X
.
Write
μ
k
⊆
k
for
the
group
def
def
def
of
roots
of
unity
of
k;
μ
k
Z
=
Hom(Q/Z,
μ
k
);
G
X
=
Gal(K
X
/K
X
);
G
k
=
Gal(k/k);
G
X
Π
X
for
the
quotient
of
G
X
determined
by
the
maximal
subextension
of
K
X
that
is
unramified
over
X
[so
Π
X
may
be
thought
of,
for
a
suitable
choice
of
basepoint,
as
the
étale
fundamental
group
of
X].
Thus,
when
k
is
an
MLF,
X
and
Π
X
are
as
in
Example
2.12.3,
(i).
Here,
for
simplicity,
we
assume
further
that
X
is
of
genus
≥
1,
and
def
write
Δ
X
=
Ker(Π
X
G
k
);
X
cp
for
the
natural
[smooth,
proper]
compactification
of
cp
X;
Δ
X
Δ
cp
X
for
the
quotient
of
Δ
X
by
the
cuspidal
inertia
groups
of
Δ
X
[so
Δ
X
may
be
naturally
identified
with
the
étale
fundamental
group,
for
a
suitable
basepoint,
of
X
cp
×
k
k];
def
M
X
=
Hom
Z
(H
2
(Δ
cp
X
,
Z),
Z)
[so
M
X
(
∼
=
Z)
is
a
cyclotome
naturally
associated
to
Δ
X
—
cf.
[AbsTopIII],
Proposition
1.4,
(ii)].
Here,
we
recall
that
the
quotient
Δ
X
Δ
cp
X
,
hence
also
the
cyclotome
M
X
,
may
be
constructed
by
means
of
a
purely
group-theoretic
algorithm
from
the
abstract
topological
group
Π
X
[cf.
[AbsTopI],
Lemma
4.5,
(v);
[IUTchI],
Remark
1.2.2,
(ii)].
Now
observe
that
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
43
the
Frobenius-like
data
that
appears
in
the
various
examples
[i.e.,
Examples
2.12.1,
2.12.2,
2.12.3]
of
mono-anabelian
transport
discussed
in
§2.12
only
involve
the
two
underlying
combinatorial
dimensions
of
[various
portions
of]
the
ind-topological
monoid
“O
k
”
of
these
examples.
That
is
to
say,
although,
for
instance,
the
étale-like
data
[i.e.,
“Π
X
”]
that
appears
in
Example
2.12.3
involves
the
relative
geometric
dimension
of
X
over
k
[i.e.,
in
the
case
where
k
is
an
MLF],
the
Frobenius-like
data
[i.e.,
“O
k
”]
that
appears
in
these
examples
does
not
involve
this
geometric
dimension
of
X
over
k.
On
the
other
hand,
in
inter-universal
Teichmüller
theory,
it
will
be
of
crucial
importance
[cf.
the
discussion
of
§3.4,
§3.6,
below;
[IUTchIII],
Remark
2.3.3]
to
consider
such
Frobenius-like
data
that
involves
the
geometric
dimension
of
X
over
k
—
i.e.,
in
more
concrete
terms,
to
consider
nonconstant
rational
functions
on
X
—
together
with
various
evaluation
operations
that
arise
by
evaluating
such
functions
on
various
“special
points”
of
X
[or
coverings
of
X].
In
fact,
the
fundamental
importance
of
such
evaluation
operations
may
also
be
seen
in
the
discussion
of
§2.14,
(i),
(ii),
(iii),
below.
In
the
remainder
of
the
present
§2.13,
we
discuss
what
is
perhaps
the
most
fundamental
example
of
cyclotomic
rigidity
and
mono-anabelian
transport
for
such
geometric
functions.
Example
2.13.1.
Monoids
of
rational
functions.
(i)
In
the
following,
we
assume
for
simplicity
that
the
field
k
is
an
NF.
Recall
that
consideration
of
the
first
Chern
class
of
a
line
bundle
of
degree
1
on
X
cp
yields
a
natural
isomorphism
∼
λ
:
μ
k
Z
→
M
X
—
cf.,
e.g.,
Example
2.6.1,
(iii);
[the
evident
NF
version
of]
[Cusp],
Proposition
1.2,
(ii).
Next,
observe
that
by
considering
the
action
of
G
X
on
the
various
N
-th
roots,
for
N
a
×
def
(
=
K
X
\
{0}),
we
obtain
a
natural
Kummer
map
positive
integer,
of
elements
of
K
X
×
κ
X
:
K
X
→
H
1
(G
X
,
μ
k
Z
)
—
which
may
be
composed
with
the
natural
isomorphism
λ
to
obtain
a
natural
embedding
×
1
κ
Gal
X
:
K
X
→
H
(G
X
,
M
X
)
—
where
we
recall
from
the
theory
of
[AbsTopIII],
§1
[cf.
[AbsTopIII],
Theorem
1.9]
that
the
Galois
group
G
X
[regarded
up
to
inner
automorphisms
that
arise
from
elements
of
Ker(G
X
Π
X
)],
together
with
the
cohomology
module
in
the
codomain
of
κ
Gal
X
,
Gal
the
image
of
κ
X
in
this
cohomology
module,
and
the
field
structure
on
the
union
K
X
(Π
X
)
Kum
of
this
image
with
{0},
may
be
constructed
group-theoretically
from
the
abstract
topological
group
Π
X
.
Write
G
X
(Π
X
)
for
“G
X
regarded
as
an
object
Shinichi
Mochizuki
44
constructed
in
this
way
from
Π
X
”;
K
X
(Π
X
)
Kum
G
X
(Π
X
)
for
the
inductive
system/limit
[which,
by
functoriality,
is
equipped
with
a
natural
ac-
tion
by
G
X
(Π
X
)]
of
the
result
of
applying
this
group-theoretic
construction
Π
X
→
K
X
(Π)
Kum
to
the
various
open
subgroups
of
G
X
(Π
X
).
×
×
(ii)
Now
let
us
regard
the
pair
“G
X
K
X
”
as
an
abstract
ind-monoid
“K
X
”
[i.e.,
inductive
system
of
monoids]
equipped
with
a
continuous
action
by
an
abstract
topological
group
“G
X
”
that
arises,
for
some
abstract
quotient
topological
group
“G
X
Π
X
”,
as
the
topological
group
“G
X
(Π
X
)”
of
(i)
[hence
is
only
well-defined
up
to
inner
automorphisms
×
that
arise
from
elements
of
Ker(G
X
Π
X
)].
Thus,
if
we
think
of
μ
k
⊆
K
X
as
the
×
subgroup
of
torsion
elements
of
the
monoid
K
X
,
then,
by
considering
the
action
of
G
X
×
on
the
various
N
-th
roots,
for
N
a
positive
integer,
of
elements
of
K
X
,
we
obtain
the
natural
Kummer
map
×
κ
X
:
K
X
→
H
1
(G
X
,
μ
k
Z
)
discussed
in
(i).
Moreover,
∼
the
{±1}-orbit
of
the
cyclotomic
rigidity
isomorphism
λ
:
μ
k
Z
→
M
X
of
×
(i)
may
be
constructed
functorially
from
the
data
“G
X
K
X
”
by
applying
the
“alternative
approach”
discussed
in
Example
2.12.1,
(iv),
as
follows
[cf.
[IUTchI],
Example
5.1,
(v);
[IUTchI],
Definition
5.2,
(vi)].
Indeed,
it
follows
formally
from
the
elementary
fact
Z
×
=
{1}
Q
>0
×
—
for
instance,
by
considering
the
various
quotients
K
X
Z
determined
by
the
discrete
valuations
of
K
X
that
arise
from
the
closed
points
of
X
cp
,
i.e.,
the
quotients
which,
at
the
level
of
Kummer
classes,
are
induced
by
restriction
to
the
various
cuspidal
inertia
groups
[cf.
the
first
display
of
[AbsTopIII],
Proposition
1.6,
(iii)]
—
that
∼
×
(Π
X
)
Kum
(
=
the
only
isomorphisms
μ
k
Z
→
M
X
that
map
the
image
of
κ
X
into
K
X
K
X
(Π
X
)
Kum
\
{0})
are
the
isomorphisms
that
belong
to
the
{±1}-orbit
of
λ.
def
As
discussed
in
Example
2.12.1,
(iv)
[cf.
also
the
discussion
of
[IUTchIII],
Remark
2.3.3,
(vii)],
this
approach
to
cyclotomic
rigidity
has
the
disadvantage
of
being
incompat-
ible
with
the
profinite
topology
of
G
X
[or
Π
X
].
(iii)
We
continue
to
use
the
notational
conventions
of
(ii).
Then
observe
that
the
functorial
construction
of
the
{±1}-orbit
of
the
cyclotomic
rigidity
isomorphism
λ
given
in
(ii)
may
be
interpreted
in
the
fashion
of
Example
2.12.2,
(i).
That
is
to
say,
observe
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
45
that
this
functorial
construction
implies
that
if
α
is
an
arbitrary
automorphism
of
the
×
data
“G
X
K
X
”,
then
either
α
or
−α
[i.e.,
the
composite
of
α
with
the
automorphism
×
×
of
the
data
“G
X
K
X
”
that
raises
elements
of
K
X
to
the
power
−1
and
acts
as
the
identity
on
G
X
]
—
but
not
both!
—
is
compatible
with
λ,
hence
also
with
κ
Gal
X
.
In
particular,
by
applying
this
observation
to
the
various
open
subgroups
of
G
X
,
one
concludes
that
one
has
a
natural
bijection
×
Aut(G
X
K
X
)
∼
→
{±1}
×
Aut(Π
X
)
×
—
where
the
first
“Aut(−)”
denotes
automorphisms
of
the
data
“G
X
K
X
”
as
de-
scribed
at
the
beginning
of
(ii);
the
second
“Aut(−)”
denotes
automorphisms
of
the
data
“Π
X
”
consisting
of
an
abstract
topological
group
[cf.
the
discussion
of
[IUTchI],
Example
5.1,
(v);
[IUTchI],
Definition
5.2,
(vi)].
Just
as
in
the
case
of
Example
2.12.2,
(i),
this
situation
may
also
be
understood
in
terms
of
the
general
framework
of
mono-
anabelian
transport
discussed
in
§2.7,
(v)
[cf.
also
Example
2.6.1,
(iii),
(iv)],
by
considering
the
commutative
diagram
×
∼
×
K
X
(Π
X
)
Kum
−→
K
X
(Π
X
)
Kum
⏐
⏐
?
⏐
Kum
⏐
Kum
×
K
X
×
∼
−→
×
K
X
def
—
where
K
X
(Π
X
)
Kum
=
K
X
(Π
X
)
Kum
\
{0};
the
horizontal
arrows
are
induced
by
×
some
given
automorphism
of
the
data
“G
X
K
X
”;
the
vertical
arrows,
which
relate
×
×
the
Frobenius-like
data
K
X
to
the
étale-like
data
K
X
(Π
X
)
Kum
,
are
the
“Kummer
isomorphisms”
determined
by
the
various
“κ
Gal
X
”
associated
to
open
subgroups
of
G
X
;
the
diagram
commutes
[cf.
“
?”]
up
to
the
action
of
a
suitable
element
∈
{±1}.
(iv)
Finally,
we
pause
to
remark
that
one
fundamental
reason
for
the
use
of
Kummer
theory
in
inter-universal
Teichmüller
theory
in
the
context
of
nonconstant
rational
functions
[i.e.,
as
in
the
discussion
of
the
present
Example
2.13.1]
lies
in
the
functoriality
of
Kummer
theory
with
respect
to
the
operation
of
evalua-
tion
of
such
functions
at
“special
points”
of
X.
That
is
to
say,
[cf.
the
discussion
of
Example
2.6.1,
(ii),
(iii);
§2.7,
(vii)]
although
there
exist
many
different
versions
—
e.g.,
versions
for
“higher-dimensional
fields”
—
of
class
field
theory,
these
versions
of
class
field
theory
do
not
satisfy
such
functoriality
properties
with
respect
to
the
operation
of
evaluation
of
functions
at
points
[cf.
the
discussion
of
§2.14,
§3.6,
§4.2,
below;
[IUTchIV],
Remark
2.3.3,
(vi)].
Shinichi
Mochizuki
46
§
2.14.
Finite
discrete
approximations
of
harmonic
analysis
Finally,
we
conclude
the
present
§2
by
pausing
to
examine
in
a
bit
more
detail
the
transition
that
was,
in
effect,
made
earlier
in
the
present
§2
in
passing
from
derivatives
[in
the
literal
sense,
as
in
the
discussion
of
§2.5]
to
Galois
groups/étale
fundamental
groups
[i.e.,
as
in
the
discussion
of
and
subsequent
to
§2.6].
This
transition
is
closely
related
to
many
of
the
ideas
of
the
[scheme-theoretic]
Hodge-Arakelov
theory
of
[HASurI],
[HASurII].
Example
2.14.1.
Finite
discrete
approximation
of
differential
calculus
on
the
real
line.
We
begin
by
recalling
that
the
differential
calculus
of
[say,
infinitely
differentiable]
functions
on
the
real
line
admits
a
finite
discrete
approximation,
namely,
by
substituting
df
(x)
dx
=
lim
δ→0
f
(x+δ)−f
(x)
δ
f
(X
+
1)
−
f
(X)
difference
operators
for
derivatives
[in
the
classical
sense].
If
d
is
a
positive
integer,
then
one
verifies
easily,
by
considering
such
difference
operators
in
the
case
of
polyno-
mial
functions
of
degree
<
d
with
coefficients
∈
Q,
that
evaluation
at
the
elements
0,
1,
.
.
.
,
d
−
1
∈
Z
⊆
Q
yields
a
natural
isomorphism
of
Q-vector
spaces
of
dimension
d
Q[X]
<d
def
=
d−1
Q
·
X
j
∼
→
d−1
Q
0
j=0
—
cf.,
e.g.,
the
discussion
of
the
well-known
classical
theory
of
Hilbert
polynomials
in
[Harts],
Chapter
I,
§7,
for
more
details.
In
fact,
it
is
not
difficult
to
compute
explicitly
the
“denominators”
necessary
to
make
this
evaluation
isomorphism
into
an
isomorphism
of
finite
free
Z-modules.
This
sort
of
“discrete
function
theory”
[cf.
also
Example
2.14.2
below]
may
be
regarded
as
the
fundamental
prototype
for
the
various
constructions
of
Hodge-Arakelov
theory.
Example
2.14.2.
Finite
discrete
approximation
of
Fourier
analysis
on
the
unit
circle.
In
the
spirit
of
the
discussion
of
Example
2.14.1,
we
recall
that
classical
function
theory
—
i.e.,
in
effect,
Fourier
analysis
—
on
the
unit
circle
S
1
admits
a
well-known
finite
discrete
approximation:
If
d
is
a(n)
[say,
for
simplicity]
odd
positive
def
integer,
so
d
∗
=
12
(d
−
1)
∈
Z,
then
one
verifies
easily
that
evaluation
of
polynomial
functions
of
degree
∈
{−d
∗
,
−d
∗
+
1,
.
.
.
,
−1,
0,
1,
d
∗
−
1,
d
∗
}
with
coefficients
∈
Z
on
the
def
multiplicative
group
scheme
G
m
=
Spec(Z[U,
U
−1
])
at
[say,
scheme-theoretic]
points
of
the
subscheme
μ
d
⊆
G
m
of
d-torsion
points
of
G
m
yields
a
natural
isomorphism
of
finite
free
Z-modules
of
rank
d
∗
d
j=−d
∗
Z
·
U
j
∼
→
O
μ
d
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
47
—
where,
by
abuse
of
notation,
we
write
O
μ
d
for
the
ring
of
global
sections
of
the
structure
sheaf
of
the
affine
scheme
μ
d
.
If
one
base-changes
via
the
natural
inclusion
Z
→
C
into
the
field
of
complex
numbers
C,
then,
when
d
is
sufficiently
“large”,
one
may
think
of
the
totality
of
these
d-torsion
points
)
⊆
S
1
exp(2πi
·
d
1
Z)
=
exp(2πi
·
d
0
),
exp(2πi
·
d
1
),
.
.
.
,
exp(2πi
·
(d−1)
d
as
a
sort
of
finite
discrete
approximation
of
S
1
and
hence,
in
particular,
of
adjacent
pairs
of
d-torsion
points
as
“tangent
vectors”
on
S
1
.
That
is
to
say,
since
[inverse
systems
of]
such
torsion
points
give
rise
to
the
étale
fundamental
group
of
G
m
×
Z
C,
it
is
precisely
this
“picture”
of
torsion
points
of
S
1
that
motivates
the
idea
that
Galois
groups/étale
fundamental
groups
should
be
regarded
as
a
sort
of
arithmetic
analogue
of
the
classical
geometric
notion
of
a
tangent
bundle
—
the
discussion
of
§2.6.
Moreover,
if
one
regards
G
m
as
the
codomain
of
a
nonzero
function
[on
some
unspecified
“space”],
then
this
very
classical
“pictorial
representation
of
a
cyclotome”
[i.e.,
of
torsion
points
of
S
1
]
also
explains,
from
a
“pictorial
point
of
view”,
the
importance
of
cyclotomes
and
Kummer
classes
in
the
discussion
of
§2.6,
§2.7.
That
is
to
say,
a
Kummer
class
of
a
function,
which,
so
to
speak,
records
the
“arithmetic
infinitesimal
motion”
in
G
m
induced,
via
the
function,
by
an
“arithmetic
in-
finitesimal
motion”
in
the
space
on
which
the
function
is
defined
may
be
thought
of
as
a
sort
of
“arithmetic
logarithmic
derivative”
of
the
function
—
a
point
of
view
that
is
consistent
with
the
usual
point
of
view
that
the
Kummer
exact
sequence
in
étale
cohomology
[i.e.,
which
induces
a
connecting
homomorphism
in
cohomology
that
computes
the
Kummer
class
of
a
function]
should
be
thought
of
as
a
sort
of
arithmetic
analogue
of
the
exponential
exact
sequence
1
→
2πi
·
Z
→
C
→
C
×
→
1
that
appears
in
the
theory
of
sheaf
cohomology
of
sheaves
of
holomorphic
functions
on
a
complex
space.
Example
2.14.3.
Finite
discrete
approximation
of
harmonic
analysis
on
complex
tori.
Examples
2.14.1
and
2.14.2
admit
a
natural
generalization
to
the
case
of
elliptic
curves.
Indeed,
let
E
be
an
elliptic
curve
over
a
field
F
of
characteristic
zero,
E
†
→
E
the
universal
extension
of
E,
η
∈
E(F
)
a
[nontrivial]
torsion
point
of
order
2,
l
=
2
a
prime
number.
Write
E[l]
⊆
E
for
the
subscheme
of
l-torsion
points,
def
L
=
O
E
(l
·
[η])
[where
“[η]”
denotes
the
effective
divisor
on
E
determined
by
η].
Here,
we
recall
that
E
†
→
E
is
an
A
1
-torsor
[so
E[l]
may
also
be
regarded
as
the
subscheme
⊆
E
†
of
l-torsion
points
of
E
†
].
In
particular,
it
makes
sense
to
speak
of
the
sections
Γ(E
†
,
L|
E
†
)
<l
⊆
Γ(E
†
,
L|
E
†
)
of
L
over
E
†
whose
relative
degree,
with
respect
to
the
48
Shinichi
Mochizuki
morphism
E
†
→
E,
is
<
l.
Then
the
simplest
version
of
the
fundamental
theorem
of
Hodge-Arakelov
theory
states
that
evaluation
at
the
subscheme
of
l-torsion
points
E[l]
⊆
E
†
yields
a
natural
isomorphism
of
F
-vector
spaces
of
dimension
l
2
Γ(E
†
,
L|
E
†
)
<l
∼
→
L|
E[l]
[cf.
[HASurI],
Theorem
A
simple
].
Moreover:
·
When
F
is
an
NF,
this
isomorphism
is
compatible,
up
to
mild
discrepancies,
with
natural
integral
structures
on
the
LHS
and
RHS
of
the
isomorphism
at
the
nonarchimedean
valuations
of
F
and
with
natural
Hermitian
metrics
on
the
LHS
and
RHS
of
the
isomorphism
at
the
archimedean
valuations
of
F
.
·
When
F
is
a
complete
discrete
valuation
field,
and
E
is
a
Tate
curve
over
F
,
with
special
fiber
isomorphic
to
G
m
,
Example
2.14.2
may
be
thought
of
as
corresponding
to
the
portion
of
the
natural
isomorphism
of
the
above
display
that
arises
from
the
“special
fiber
of
E”,
while
Example
2.14.1
may
be
thought
of
as
corresponding
to
the
portion
of
the
natural
isomorphism
of
the
above
display
that
arises
from
the
“special
fiber
of
the
relative
dimension
of
the
morphism
E
†
→
E”.
·
When
E
is
a
Tate
curve,
the
isomorphism,
over
F
,
of
the
above
display
may
be
interpreted
as
a
result
concerning
the
invertibility
of
the
matrix
determined
by
the
values
at
the
l-torsion
points
of
certain
theta
functions
associated
to
the
Tate
curve
and
their
derivatives
of
order
<
l
[cf.,
§3.4,
(iii),
below;
§3.6,
(ii),
below;
[Fsk],
§2.5;
[EtTh],
Proposition
1.4,
for
a
review
of
the
series
repre-
sentation
of
such
theta
functions].
·
When
F
is
an
arbitrary
field,
the
isomorphism
of
the
above
display
may
be
thought
of
as
a
sort
of
discrete
polynomial
version
of
the
Gaussian
inte-
√
2
∞
gral
−∞
e
−x
dx
=
π,
i.e.,
in
the
spirit
of
the
discussion
of
Examples
2.14.1
and
2.14.2,
above.
·
When
F
is
an
NF,
the
isomorphism
of
the
above
display
may
be
thought
of
as
a
sort
of
discrete
globalized
version
of
the
harmonic
analysis
involv-
ing
“∂,
∂,
Green’s
functions,
etc.”
that
appears
at
archimedean
valuations
in
classical
Arakelov
theory.
This
is
the
reason
for
the
appearance
of
the
word
“Arakelov”
in
the
term
“Hodge-Arakelov
theory”.
From
this
point
of
view,
the
computation
of
the
discrepancy
between
natural
integral
structures/metrics
on
the
LHS
and
RHS
of
the
isomorphism
of
the
above
display
may
be
thought
of
as
a
sort
of
computation
of
analytic
torsion
—
a
point
of
view
that
in
some
sense
foreshadows
the
interpretation
[cf.
the
discussion
of
§3.9,
(iii),
below]
of
inter-universal
Teichmüller
theory
as
the
computation
of
a
sort
of
global
arithmetic/Galois-theoretic
form
of
analytic
torsion.
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
49
·
The
isomorphism
of
the
above
display
may
also
be
thought
of
as
a
sort
of
global
(“function-theoretic”!)
arithmetic
version
of
the
(“linear”!)
comparison
isomorphisms
that
occur
in
complex
or
p-adic
Hodge
theory.
[That
is
to
say,
the
LHS
and
RHS
of
the
isomorphism
of
the
above
display
correspond,
respectively,
to
the
“de
Rham”
and
“étale”
sides
of
comparison
iso-
morphisms
in
p-adic
Hodge
theory.]
This
is
the
reason
for
the
appearance
of
the
word
“Hodge”
in
the
term
“Hodge-Arakelov
theory”.
This
point
of
view
gives
rise
to
a
natural
definition
for
a
sort
of
arithmetic
version
of
the
Kodaira-
Spencer
morphism
discussed
in
§2.9,
in
which
Galois
groups
play
the
role
played
by
tangent
bundles
in
the
classical
version
of
the
Kodaira-Spencer
morphism
reviewed
in
§2.9
[cf.
[HASurI],
§1.4].
When
E
is
a
Tate
curve
over
a
complete
discrete
valuation
field
F
,
this
arithmetic
Kodaira-Spencer
morphism
essentially
coincides,
when
formulated
properly,
with
the
classical
Kodaira-Spencer
morphism
reviewed
in
§2.9
[cf.
[HASurII],
Corollary
3.6].
·
Relative
to
the
point
of
view
of
“filtered
crystals”
[e.g.,
vector
bundles
equipped
with
a
connection
and
filtration
—
cf.
the
data
(E,
∇
E
,
ω
E
⊆
E)
of
§2.9],
the
isomorphism
of
the
above
display
may
be
thought
of
as
a
sort
of
discrete
Galois-theoretic
version
of
the
“crystalline
theta
object”
[cf.
[HASurII],
§2.3;
the
remainder
of
[HASurII],
§2],
i.e.,
in
essence,
the
“nonlinear
filtered
crystal”
constituted
by
the
universal
extension
E
†
equipped
with
the
ample
line
bundle
L|
E
†
,
the
natural
crystal
structure
on
(E
†
,
L|
E
†
),
and
the
“filtration”
constituted
by
the
morphism
E
†
→
E.
We
refer
to
[HASurI],
[HASurII],
for
more
details
concerning
the
ideas
just
dis-
cussed.
§
3.
Multiradiality:
an
abstract
analogue
of
parallel
transport
§
3.1.
The
notion
of
multiradiality
So
far,
in
§2,
we
have
discussed
various
generalities
concerning
arithmetic
changes
of
coordinates
[cf.
§2.10;
the
analogy
discussed
in
§2.2,
§2.5,
and
§2.7
with
the
classical
theory
of
§1.4
and
§1.5],
which
are
applied
in
effect
to
the
two
underlying
combina-
torial
dimensions
of
a
ring
such
as
an
MLF
or
an
NF
[cf.
§2.7,
(vii);
§2.11;
§2.12],
and
the
approach
to
computing
the
effect
of
such
arithmetic
changes
of
coordinates
—
i.e.,
in
the
form
of
Kummer-detachment
indeterminacies
or
étale-transport
in-
determinacies
[cf.
§2.7,
(vi);
§2.9]
—
by
means
of
the
technique
of
mono-anabelian
transport
[cf.
§2.7,
(v)].
By
contrast,
in
the
present
§3,
we
turn
to
the
issue
of
con-
sidering
the
particular
arithmetic
changes
of
coordinates
that
are
of
interest
in
the
50
Shinichi
Mochizuki
context
of
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
§2.1,
§2.3,
§2.4].
Many
aspects
of
these
particular
arithmetic
changes
of
coordinates
are
highly
reminis-
cent
of
the
change
of
coordinates
discussed
in
§1.6
from
planar
cartesian
to
polar
coordinates.
In
some
sense,
the
central
notion
that
underlies
the
abstract
combinatorial
analogue,
i.e.,
that
is
developed
in
inter-universal
Teichmüller
theory,
of
this
change
of
coordinates
from
planar
cartesian
to
polar
coordinates
is
the
notion
of
multiradiality.
(i)
Types
of
mathematical
objects:
In
the
following
discussion,
we
shall
often
speak
of
“types
of
mathematical
objects”,
i.e.,
such
as
groups,
rings,
topological
spaces
equipped
with
some
additional
structure,
schemes,
etc.
This
notion
of
a
“type
of
mathematical
object”
is
formalized
in
[IUTchIV],
§3,
by
introducing
the
notion
of
a
“species”.
On
the
other
hand,
the
details
of
this
formalization
are
not
so
important
for
the
following
discussion
of
the
notion
of
multiradiality.
A
“type
of
mathematical
object”
determines
an
associated
category
consisting
of
mathematical
objects
of
this
type
—
i.e.,
in
a
given
universe,
or
model
of
set
theory
—
and
morphisms
between
such
mathematical
objects.
On
the
other
hand,
in
general,
the
structure
of
this
associated
category
[i.e.,
as
an
abstract
category!]
contains
considerably
less
information
than
the
information
that
determines
the
“type
of
mathematical
object”
that
one
started
with.
For
instance,
if
p
is
a
prime
number,
then
the
“type
of
mathematical
object”
given
by
rings
isomorphic
to
Z/pZ
[and
ring
homomorphisms]
yields
a
category
whose
equivalence
class
as
an
abstract
category
is
manifestly
independent
of
the
prime
number
p.
(ii)
Radial
environments:
A
radial
environment
consists
of
a
triple.
The
first
member
of
this
triple
is
a
specific
“type
of
mathematical
object”
that
is
referred
to
as
radial
data.
The
second
member
of
this
triple
is
a
specific
“type
of
mathematical
object”
that
is
referred
to
as
coric
data.
The
third
member
of
this
triple
is
a
functorial
algorithm
that
inputs
radial
data
and
outputs
coric
data;
this
algorithm
is
referred
to
as
radial,
while
the
resulting
functor
from
the
category
of
radial
data
to
the
category
of
coric
data
is
referred
to
as
the
radial
functor
of
the
radial
environment.
We
would
like
to
think
of
the
coric
data
as
a
sort
of
“underlying
structure”
of
the
“finer
structure”
constituted
by
the
radial
data
and
of
the
radial
algorithm
as
an
algorithm
that
forgets
this
“finer
structure”,
i.e.,
an
algorithm
that
assigns
to
a
collection
of
radial
data
the
collection
of
underlying
coric
data
of
this
given
collection
of
radial
data.
We
refer
to
[IUTchII],
Example
1.7,
(i),
(ii),
for
more
details.
(iii)
Multiradiality
and
uniradiality:
A
radial
environment
is
called
multira-
dial
if
its
associated
radial
functor
is
full.
A
radial
environment
is
called
uniradial
if
its
associated
radial
functor
is
not
full.
One
important
consequence
of
the
condition
of
multiradiality
is
the
following
switching
property:
Consider
the
category
of
objects
consisting
of
an
ordered
pair
of
collections
of
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
51
radial
data,
together
with
an
isomorphism
between
the
associated
collections
of
underlying
coric
data
[and
morphisms
defined
in
the
evident
way].
Observe
that
this
category
admits
a
switching
functor
[from
the
category
to
itself]
that
assigns
to
an
object
of
the
category
the
object
obtained
by
switching
the
two
collections
of
radial
data
of
the
given
object
and
replaces
the
isomorphism
between
associated
collections
of
underlying
coric
data
by
the
inverse
to
this
isomorphism.
Then
multiradiality
implies
that
the
switching
functor
preserves
the
isomorphism
classes
of
objects.
Indeed,
one
verifies
immediately
that
multiradiality
is
in
fact
equivalent
to
the
property
that
any
object
of
the
category
discussed
in
the
above
display
is,
in
fact,
isomorphic
to
a
“diagonal
object”,
i.e.,
an
object
given
by
considering
an
ordered
pair
of
copies
of
a
given
collection
of
radial
data,
together
with
the
identity
isomorphism
between
the
associated
collections
of
underlying
coric
data
—
cf.
the
illustration
of
Fig.
3.1
below.
We
refer
to
[IUTchII],
Example
1.7,
(ii),
(iii),
for
more
details.
(iv)
Analogy
with
the
Grothendieck
definition
of
a
connection:
Thus,
in
summary,
multiradiality
concerns
the
issue
of
comparison
between
two
collections
of
radial
data
that
share
a
common
collection
of
underlying
coric
data.
We
shall
often
think
of
this
sort
of
comparison
as
a
comparison
between
two
“holomor-
phic
structures”
that
share
a
common
“underlying
real
analytic
structure”
[cf.
the
examples
discussed
in
§3.2
below].
Note
that
multiradiality
may
be
thought
of
as
a
sort
of
abstract
analogue
of
the
notion
of
“parallel
transport”
or,
alternatively,
the
Grothendieck
definition
of
a
connection
[cf.
the
discussion
of
[IUTchII],
Remark
1.7.1].
That
is
to
say,
given
a
scheme
X
over
a
scheme
S,
the
Grothendieck
definition
of
a
connection
on
an
object
E
over
X
consists
of
an
isomorphism
between
the
fibers
of
E
at
two
distinct
—
but
infinitesimally
close!
—
points
of
X
that
map
to
the
same
point
of
S.
Thus,
one
may
think
of
the
fullness
condition
of
multiradiality
as
the
condition
that
there
exist
a
sort
of
parallel
transport
isomorphism
between
two
collections
of
radial
data
[i.e.,
corresponding
to
two
“fibers”]
that
lifts
a
given
isomorphism
between
collections
of
underlying
coric
data
[i.e.,
corresponding
to
a
path
between
the
points
over
which
the
two
fibers
lie].
The
indeterminacy
in
the
choice
of
such
a
lifting
may
then
be
thought
of,
relative
to
this
analogy
with
parallel
transport,
as
a
sort
of
“monodromy”
associated
to
the
multiradial
environment.
(v)
The
Kodaira-Spencer
morphism
via
multiradiality:
The
classical
ap-
proach
to
proving
the
geometric
version
of
the
Szpiro
Conjecture
by
means
of
the
Kodaira-Spencer
morphism
was
reviewed
in
§2.9.
Here,
we
observe
that
this
argument
involving
the
Kodaira-Spencer
morphism
may
be
formulated
in
a
way
that
Shinichi
Mochizuki
52
permutable?
radial
data
radial
algorithm
radial
data
radial
algorithm
coric
data
Fig.
3.1:
Multiradiality
vs.
uniradiality
·
renders
explicit
the
analogy
discussed
in
(iv)
above
between
multiradiality
and
connections,
and,
moreover,
·
renders
explicit
the
relationship
between
this
classical
argument
involving
the
Kodaira-Spencer
morphism
and
the
approach
taken
in
inter-universal
Te-
ichmüller
theory,
that
is
to
say,
as
a
sort
of
“limiting
case”
or
“degenerate
version”
of
the
argument
[sketched
in
the
Introduction
to
the
present
paper
—
cf.
also
the
discussion
of
§2.3,
§2.4]
involving
multiplication
of
the
height
“h”
by
a
factor
“N
”,
in
the
limit
“N
→
∞”
[in
which
case
comparison
be-
tween
“h”
and
“N
·
h”,
or
equivalently,
between
“h”
and
“
N
1
·
h”,
becomes
a
comparison
between
“h”
and
“0”].
This
formulation
may
be
broken
down
into
steps,
as
follows.
Let
S
log
be
as
in
§2.9,
L
a
line
bundle
on
S.
Suppose
that
we
are
interested
in
bounding
deg(L)
[i.e.,
bounding
the
degree
of
L
from
above].
Then:
(1
KS
)
Write
p
1
,
p
2
:
S
×
S
→
S
for
the
natural
projections
from
the
direct
product
S
×
S
to
the
first
and
second
factors.
Suppose
that
we
are
given
an
isomorphism
∼
p
∗
1
L
→
p
∗
2
L
of
line
bundles
on
S
×
S
between
the
pull-backs
of
L
via
p
1
,
p
2
.
Then
one
verifies
immediately,
by
restricting
to
various
fibers
of
the
direct
product
S
×
S,
that
the
existence
of
such
an
isomorphism
implies
that
deg(L)
=
0
[hence
that
deg(L)
is
bounded],
as
desired.
(2
∼
KS
)
Write
S
δ
for
the
first
infinitesimal
neighborhood
of
the
diagonal
(S
→
)
Δ
S
⊆
S
×
S.
Suppose
that
we
are
given
an
isomorphism
∼
p
∗
1
L|
S
δ
→
p
∗
2
L|
S
δ
of
line
bundles
on
S
δ
between
the
restrictions
to
S
δ
of
the
two
pull-backs
via
p
1
,
p
2
of
L.
Since
S
is
proper
[so
any
automorphism
of
the
line
bundle
L
on
S
is
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
53
given
by
multiplication
by
a
nonzero
complex
number],
one
verifies
immediately
that
an
isomorphism
as
in
the
above
display
may
be
thought
of
as
a
connection,
in
the
sense
of
Grothendieck,
on
L
[cf.
the
discussion
of
(iv)
above!].
In
particular,
since
the
base
field
C
is
of
characteristic
zero,
we
thus
conclude
again
[from
the
elementary
theory
of
de
Rham-theoretic
first
Chern
classes
of
line
bundles
on
curves]
that
deg(L)
=
0
[and
hence
that
deg(L)
is
bounded],
as
desired.
(3
KS
)
Let
F
be
a
rank
two
vector
bundle
that
admits
an
exact
sequence
0
→
L
→
F
→
L
−1
→
0
of
vector
bundles
on
S.
Thus,
one
may
think
of
F
as
a
container
for
L.
Write
S
δ
log
for
the
first
infinitesimal
neighborhood
of
the
“logarithmic
diagonal”
∼
(S
log
→
)
Δ
S
log
⊆
S
log
×
S
log
.
Next,
suppose
that
we
are
given
an
isomorphism
∼
p
∗
1
F|
S
log
→
p
∗
2
F|
S
log
δ
δ
of
vector
bundles
on
[the
underlying
scheme
of]
S
δ
log
between
the
restrictions
to
S
δ
log
of
the
pull-backs
via
p
1
,
p
2
of
F.
[Thus,
such
an
isomorphism
arises,
for
instance,
from
a
logarithmic
connection
on
F.]
Suppose,
moreover,
that
this
isomorphism
has
nilpotent
monodromy,
i.e.,
that
the
restriction
of
this
isomorphism
to
each
of
the
cusps
of
S
log
differs
from
multiplication
by
a
nonzero
complex
number
by
a
nilpotent
endomorphism
of
the
fiber
of
F
at
the
cusp
under
consideration.
Thus,
we
obtain
two
inclusions
∼
p
∗
1
L|
S
log
→
p
∗
1
F|
S
log
→
p
∗
2
F|
S
log
←
p
∗
2
L|
S
log
δ
δ
δ
δ
∼
KS
[where
the
“
→
”
is
the
isomorphism
of
the
first
display
of
the
present
(3
)]
of
line
bundles
into
a
rank
two
vector
bundle
over
S
δ
log
;
one
verifies
immediately
that
the
images
of
these
two
inclusions
coincide
over
the
diagonal
Δ
S
log
⊆
S
δ
log
.
That
is
to
say,
∼
the
isomorphism
p
∗
1
F|
S
log
→
p
∗
2
F|
S
log
allows
one
to
use
F
as
a
con-
δ
δ
tainer
for
L
to
compare
the
discrepancy
between
the
two
[restrictions
to
S
δ
log
of]
pull-backs
p
∗
1
L|
S
log
,
p
∗
2
L|
S
log
.
δ
(4
KS
δ
∼
)
Suppose
that
the
images
in
p
∗
1
F|
S
log
→
p
∗
2
F|
S
log
of
the
two
inclusions
in
the
KS
δ
δ
second
display
of
(3
)
coincide.
Then
[since
S
is
proper
—
cf.
the
argument
∼
KS
in
(2
)]
the
resulting
isomorphism
p
∗
1
L|
S
log
→
p
∗
2
L|
S
log
may
be
thought
of
as
δ
δ
a
logarithmic
connection
on
L
with
nilpotent
monodromy,
i.e.,
[since
L
is
of
rank
one!]
a
connection
[without
logarithmic
poles!]
on
L.
In
particular,
we
are
in
the
KS
situation
of
(2
),
so
we
may
conclude
again
that
deg(L)
=
0
[and
hence
that
deg(L)
is
bounded],
as
desired.
Shinichi
Mochizuki
54
(5
KS
∼
)
In
general,
of
course,
the
images
in
p
∗
1
F|
S
log
→
p
∗
2
F|
S
log
of
the
two
inclusions
δ
KS
δ
in
the
second
display
of
(3
)
will
not
coincide.
On
the
other
hand,
in
this
case
[i.e.,
in
which
the
images
of
the
two
inclusions
do
not
coincide],
one
may
consider
KS
[cf.
the
diagram
in
the
second
display
of
(3
)]
the
composite
∼
p
∗
1
L|
S
log
→
p
∗
1
F|
S
log
→
p
∗
2
F|
S
log
p
∗
2
L
−1
|
S
log
δ
δ
δ
δ
[where
the
“”
is
the
restriction
to
S
δ
log
of
the
given
surjection
F
L
−1
],
whose
restriction
to
Δ
S
log
⊆
S
δ
log
vanishes,
hence
determines
a
nonzero
morphism
of
line
bundles
on
S
L
→
ω
S
log
/C
⊗
L
−1
[where
we
recall
that
the
ideal
sheaf
defining
the
closed
[log]
subscheme
Δ
S
log
⊆
S
δ
log
may
be
naturally
identified
with
the
push-forward,
via
the
natural
inclusion
Δ
S
log
→
S
δ
log
,
of
the
sheaf
of
logarithmic
differentials
ω
S
log
/C
].
Now
one
verifies
immediately
that,
if
one
takes
F
to
be
the
vector
bundle
“E”
of
§2.9,
equipped
KS
with
the
isomorphism
as
in
the
first
display
of
(3
)
arising
from
the
logarithmic
connection
∇
E
,
and
L
to
be
the
subbundle
“ω
E
⊆
E”
of
§2.9,
then
the
nonzero
mor-
phism
of
the
above
display
may
be
identified
with
the
Kodaira-Spencer
morphism
ω
E
→
τ
E
⊗
O
S
ω
S
log
/C
discussed
in
§2.9.
Thus,
in
summary,
the
Kodaira-Spencer
morphism
may
be
thought
of
as
a
measure
of
the
discrepancy
that
arises
when
one
fixes
the
“ω
E
”
on
one
factor
of
S
and
compares
it
with
the
“ω
E
”
on
a
distinct,
“alien”
factor
of
S
by
means
of
the
common
container
“E”,
which
is
equipped
with
a
KS
connection
“∇
E
”
[i.e.,
an
isomorphism
as
in
the
first
display
of
(3
)].
When
formulated
in
this
way,
the
Kodaira-Spencer
morphism
becomes
manifestly
analogous
to
the
approach
sketched
in
the
Introduction
to
the
present
paper
[cf.
also
the
discussion
of
§2.3,
§2.4]
to
bounding
heights
of
elliptic
curves
[cf.
the
discussion
of
§3.7,
(ii),
(iv),
below]
by
applying
a
suitable
multiradiality
property
[cf.
the
discussion
of
§3.7,
(i),
below],
i.e.,
[in
the
language
of
the
Introduction]
a
“license
to
confuse”.
[Here,
we
note
that,
relative
to
the
analogy
with
inter-universal
Teichmüller
theory,
the
KS
KS
situation
that
arises
in
(2
),
(4
)
corresponds
to
the
[unusual!]
situation
in
which
there
actually
exists
a
“global
multiplicative
subspace”
—
cf.
the
discussion
of
§2.3.]
Finally,
we
remark
in
passing
that
the
crystalline
theta
object
referred
to
in
the
discussion
of
Example
2.14.3
may
be
thought
of
as
a
sort
of
intermediate
stage
between
KS
the
situation
discussed
in
(5
)
and
the
situation
that
is
ultimately
considered
in
inter-
universal
Teichmüller
theory.
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
∼
C
→
R
2
radial
algorithm
55
∼
C
→
R
2
radial
algorithm
GL
2
(R)
R
2
Fig.
3.2:
The
uniradiality
of
complex
holomorphic
structures
§
3.2.
Fundamental
examples
of
multiradiality
The
following
examples
may
be
thought
of
as
fundamental
prototypes
of
the
phenomenon
of
multiradiality.
Example
3.2.1.
Complex
holomorphic
structures
on
two-dimensional
real
vector
spaces.
(i)
Consider
the
radial
environment
in
which
the
radial
data
is
given
by
one-
dimensional
C-vector
spaces
[and
isomorphisms
between
such
data],
the
coric
data
is
given
by
two-dimensional
R-vector
spaces
[and
isomorphisms
between
such
data],
and
the
radial
algorithm
assigns
to
a
one-dimensional
C-vector
space
the
associated
underlying
R-vector
space.
Then
one
verifies
immediately
that
this
radial
environment,
shown
in
Fig.
3.2
above,
is
uniradial
[cf.
[Pano],
Figs.
2.2,
2.3].
(ii)
If
V
is
a
two-dimensional
real
vector
space,
then
write
End(V
)
for
the
R-algebra
of
R-linear
endomorphisms
of
V
and
GL(V
)
for
the
group
of
invertible
elements
of
End(V
).
Observe
that
if
V
is
a
two-dimensional
real
vector
space,
then
a
complex
—
i.e.,
“holomorphic”
—
structure
on
V
may
be
thought
of
as
a
homomorphism
of
R-algebras
C
→
End(V
).
In
particular,
it
makes
sense
to
speak
of
a
GL-orbit
of
complex
structures
on
V
,
i.e.,
the
set
of
GL(V
)-conjugates
of
some
such
homomorphism.
Now
consider
the
radial
environment
in
which
a
collection
of
radial
data
consists
of
a
two-
dimensional
R-vector
space
equipped
with
a
GL-orbit
of
complex
structures
[and
the
morphisms
between
such
data
are
taken
to
be
the
isomorphisms
between
such
data],
the
coric
data
is
the
same
as
the
coric
data
of
(i),
and
the
radial
algorithm
assigns
to
a
two-dimensional
R-vector
space
equipped
with
a
GL-orbit
of
complex
structures
the
associated
underlying
R-vector
space.
Then
one
verifies
immediately
that
this
radial
environment,
shown
in
Fig.
3.3
below,
is
[“tautologically”!]
multiradial
[cf.
[Pano],
Figs.
2.2,
2.3].
(iii)
The
examples
of
radial
environments
discussed
in
(i),
(ii)
are
particularly
of
interest
in
the
context
of
inter-universal
Teichmüller
theory
in
light
of
the
relationship
Shinichi
Mochizuki
56
GL
2
(R)
C
→
R
2
radial
algorithm
GL
2
(R)
C
→
R
2
∼
∼
radial
algorithm
GL
2
(R)
R
2
Fig.
3.3:
The
multiradiality
of
GL
2
(R)-orbits
of
complex
holomorphic
structures
between
complex
holomorphic
structures
as
discussed
in
(i),
(ii)
and
the
geometry
of
the
upper
half-plane.
That
is
to
say,
if,
in
the
notation
of
(ii),
we
write
GL(V
)
=
GL
−
(V
)
for
the
decomposition
of
GL(V
)
determined
by
considering
the
GL
+
(V
)
sign
of
the
determinant
of
an
R-linear
automorphism
of
V
,
then
the
space
of
moduli
of
complex
holomorphic
structures
on
V
[i.e.,
the
set
of
GL(V
)-conjugates
of
a
particular
homomorphism
of
R-algebras
C
→
End(V
)]
may
be
identified
GL(V
)/C
×
∼
→
GL
+
(V
)/C
×
GL
−
(V
)/C
×
∼
→
H
+
H
−
in
a
natural
way
with
the
disjoint
union
of
the
upper
[i.e.,
H
+
]
and
lower
[i.e.,
H
−
]
half-planes.
This
observation
is
reminiscent
of
the
deep
connections
between
inter-
universal
Teichmüller
theory
and
the
hyperbolic
geometry
of
the
upper
half-plane,
as
discussed
in
[BogIUT]
[cf.
also
the
discussion
of
§2.4,
as
well
as
of
§3.10,
(vi);
§4.1,
(i);
§4.3,
(iii),
of
the
present
paper].
This
circle
of
ideas
is
also
of
interest,
in
the
context
of
inter-universal
Teichmüller
theory,
in
the
sense
that
it
is
reminiscent
of
the
natural
bijection
∼
C
×
\GL
+
(V
)/C
×
→
[0,
1)
t
0
→
t−1
t+1
0
1
[where
R
t
≥
1]
between
the
space
of
double
cosets
on
the
left
and
the
semi-closed
interval
[0,
1)
on
the
right,
i.e.,
a
bijection
that
is
usually
interpreted
in
classical
complex
Teichmüller
theory
as
the
map
that
assigns
to
a
deformation
of
complex
structure
the
dilation
∈
[0,
1)
associated
to
this
deformation
[cf.
[QuCnf],
Proposition
A.1,
(ii)].
Example
3.2.2.
Arithmetic
fundamental
groups
of
hyperbolic
curves
of
strictly
Belyi
type
over
mixed-characteristic
local
fields.
(i)
Let
X
→
Spec(k)
and
Π
X
G
k
be
as
in
Example
2.12.3,
(i).
Consider
the
radial
environment
in
which
the
radial
data
is
given
by
topological
groups
Π
that
“just
happen
to
be”
abstractly
isomorphic
as
topological
groups
to
Π
X
[and
isomorphisms
between
topological
groups],
the
coric
data
is
given
by
topological
groups
G
that
“just
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
Π
G
radial
algorithm
57
Π
G
radial
algorithm
Aut(G)
G
Fig.
3.4:
The
uniradiality
of
local
arithmetic
holomorphic
structures
Aut(G)
Π
G
radial
algorithm
Aut(G)
Π
G
radial
algorithm
Aut(G)
G
Fig.
3.5:
The
multiradiality
of
Aut(G)-orbits
of
local
arithmetic
holomorphic
structures
happen
to
be”
abstractly
isomorphic
as
topological
groups
to
G
k
[and
isomorphisms
between
topological
groups],
and
the
radial
algorithm
assigns
to
a
topological
group
Π
the
quotient
group
Π
G
that
corresponds
to
the
[group-theoretically
constructible!
—
cf.
the
discussion
of
Example
2.12.3,
(i)]
quotient
Π
X
G
k
.
Then
one
verifies
immediately
[cf.
the
two
displays
of
Example
2.12.3,
(i)!]
that
this
radial
environment,
shown
in
Fig.
3.4
above,
is
uniradial.
(ii)
In
the
remainder
of
the
present
(ii),
we
apply
the
notation
“Aut(−)”
to
denote
the
group
of
automorphisms
of
the
topological
group
in
parentheses.
Consider
the
radial
environment
in
which
the
radial
data
is
given
by
triples
(Π,
G,
α)
—
where
Π
is
a
topological
group
as
in
the
radial
data
of
(i),
G
is
a
topological
group
as
in
the
coric
data
of
(i),
and
α
is
an
Aut(G)-orbit
of
isomorphisms
between
G
and
the
quotient
of
Π
that
corresponds
to
the
[group-theoretically
constructible!]
quotient
Π
X
G
k
—
[and
isomorphisms
between
such
triples],
the
coric
data
is
the
same
as
the
coric
data
of
(i),
and
the
radial
algorithm
assigns
to
a
triple
(Π,
G,
α)
the
topological
group
G.
Then
one
verifies
immediately
that
this
radial
environment,
shown
in
Fig.
3.5
above,
is
[“tautologically”!]
multiradial
[cf.
[IUTchII],
Example
1.8,
(i)].
Shinichi
Mochizuki
58
§
3.3.
The
log-theta-lattice:
Θ
±ell
N
F
-Hodge
theaters,
log-links,
Θ-links
The
fundamental
stage
on
which
the
constructions
of
inter-universal
Teichmüller
theory
are
performed
is
referred
to
as
the
log-theta-lattice.
(i)
Initial
Θ-data:
The
log-theta-lattice
is
completely
determined,
up
to
isomor-
phism,
once
one
fixes
a
collection
of
initial
Θ-data.
Roughly
speaking,
this
data
consists
of
·
an
elliptic
curve
E
F
over
a
number
field
F
,
·
an
algebraic
closure
F
of
F
,
·
a
prime
number
l
≥
5,
·
a
collection
of
valuations
V
of
a
certain
subfield
K
⊆
F
,
and
·
a
collection
of
valuations
V
bad
mod
of
a
certain
subfield
F
mod
⊆
F
that
satisfy
certain
technical
conditions
—
cf.
[IUTchI],
Definition
3.1,
for
more
details.
Here,
we
write
F
mod
⊆
F
for
the
field
of
moduli
of
E
F
,
i.e.,
the
field
extension
of
Q
obtained
by
adjoining
the
j-invariant
of
E
F
;
K
⊆
F
for
the
extension
field
of
F
generated
by
the
fields
of
definition
of
the
l-torsion
points
of
E
F
;
X
F
⊆
E
F
for
the
once-punctured
elliptic
curve
obtained
by
removing
the
origin
from
E
F
;
and
X
F
→
C
F
for
the
hyperbolic
orbicurve
obtained
by
forming
the
stack-theoretic
quotient
of
X
F
by
the
natural
action
of
{±1}.
Also,
in
the
following,
we
shall
write
V(−)
for
the
set
of
all
[nonarchimedean
and
archimedean]
valuations
of
an
NF
“(−)”
and
append
a
subscripted
element
∈
V(−)
to
the
NF
to
denote
the
completion
of
the
NF
at
the
element
∈
V(−).
We
assume
further
that
the
following
conditions
are
satisfied
[cf.
[IUTchI],
Definition
3.1,
for
more
details]:
·
F
is
Galois
over
F
mod
of
degree
prime
to
l
and
contains
the
fields
of
definition
of
the
2·3-torsion
points
of
E
F
;
·
the
image
of
the
natural
inclusion
Gal(K/F
)
→
GL
2
(F
l
)
[well-defined
up
to
composition
with
an
inner
automorphism]
contains
SL
2
(F
l
);
·
E
F
has
stable
reduction
at
all
of
the
nonarchimedean
valuations
of
F
;
def
·
C
K
=
C
F
×
F
K
is
a
K-core,
i.e.,
does
not
admit
a
finite
étale
covering
that
is
isomorphic
to
a
finite
étale
covering
of
a
Shimura
curve
[cf.
[CanLift],
Remarks
2.1.1,
2.1.2];
this
condition
implies
that
there
exists
a
unique
model
C
F
mod
of
C
F
over
F
mod
[cf.
the
discussion
of
[IUTchI],
Remark
3.1.7,
(i)];
·
V
⊆
V(K)
is
a
subset
such
that
the
natural
inclusion
F
mod
⊆
F
⊆
K
induces
∼
def
a
bijection
V
→
V
mod
between
V
and
the
set
V
mod
=
V(F
mod
);
·
V
bad
mod
⊆
V
mod
is
a
nonempty
set
of
nonarchimedean
valuations
of
odd
residue
characteristic
over
which
E
F
has
bad
[i.e.,
multiplicative]
reduction,
that
is
to
say,
roughly
speaking,
the
subset
of
the
set
of
valuations
where
E
F
has
bad
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
59
multiplicative
reduction
that
will
be
“of
interest”
to
us
in
the
context
of
the
constructions
of
inter-universal
Teichmüller
theory.
The
above
conditions
in
fact
imply
that
K
is
Galois
over
F
mod
[cf.
[IUTchI],
Remark
3.1.5].
We
shall
write
def
V
bad
=
V
bad
mod
×
V
mod
V
⊆
V,
def
bad
V
good
mod
=
V
mod
\
V
mod
,
def
V
good
=
V
\
V
bad
and
apply
the
superscripts
“non”
and
“arc”
to
V,
V
mod
,
and
V(−)
to
denote
the
subsets
of
nonarchimedean
and
archimedean
valuations,
respectively.
The
data
listed
above
determines,
up
to
K-isomorphism
[cf.
[IUTchI],
Remark
3.1.3],
a
finite
étale
covering
C
K
→
C
K
of
degree
l
such
that
the
base-changed
covering
X
K
def
=
C
K
×
C
F
X
F
→
X
K
def
=
X
F
×
F
K
arises
from
a
rank
one
quotient
E
K
[l]
Q
(
∼
=
Z/lZ)
of
the
module
E
K
[l]
of
l-torsion
def
points
of
E
K
(K)
[where
we
write
E
K
=
E
F
×
F
K]
which,
at
v
∈
V
bad
,
restricts
to
the
quotient
arising
from
coverings
of
the
dual
graph
of
the
special
fiber.
(ii)
The
log-theta-lattice:
The
log-theta-lattice,
various
versions
of
which
are
defined
in
[IUTchIII]
[cf.
[IUTchIII],
Definitions
1.4;
3.8,
(iii)],
is
a
[highly
noncom-
mutative!]
two-dimensional
diagram
that
consists
of
three
types
of
components,
namely,
•’s,
↑’s,
and
→’s
[cf.
the
portion
of
Fig.
3.6
below
that
lies
to
the
left
of
the
“⊇”].
Each
“•”
in
Fig.
3.6
represents
a
Θ
±ell
N
F
-Hodge
theater,
which
may
be
thought
of
as
a
sort
of
miniature
model
of
the
conventional
arithmetic
geometry
surrounding
the
given
initial
Θ-data.
Each
vertical
arrow
“↑”
in
Fig.
3.6
represents
a
log-link,
i.e.,
a
certain
type
of
gluing
between
various
portions
of
the
Θ
±ell
N
F
-
Hodge
theaters
that
constitute
the
domain
and
codomain
of
the
arrow.
Each
horizontal
arrow
“→”
in
Fig.
3.6
represents
a
Θ-link
[various
versions
of
which
are
defined
in
[IUTchI],
[IUTchII],
[IUTchIII]],
i.e.,
another
type
of
gluing
between
various
portions
of
the
Θ
±ell
N
F
-Hodge
theaters
that
constitute
the
domain
and
codomain
of
the
arrow.
The
portion
of
the
log-theta-lattice
that
is
ultimately
actually
used
to
prove
the
main
results
of
inter-universal
Teichmüller
theory
is
shown
in
the
portion
of
Fig.
3.6
—
i.e.,
a
sort
of
“infinite
letter
H”
—
that
lies
to
the
right
of
the
“⊇”.
On
the
other
hand,
the
significance
of
considering
the
entire
log-theta-lattice
may
be
seen
in
the
fact
that
—
unlike
the
portion
of
Fig.
3.6
that
lies
to
the
right
of
the
“⊇”!
—
the
[entire]
log-theta-lattice
is
symmetric,
up
to
unique
isomorphism,
with
respect
to
arbitrary
horizontal
and
vertical
translations.
Various
objects
constructed
from
the
•’s
of
the
log-theta-lattice
will
be
referred
to
as
horizontally
coric
if
they
are
invariant
with
respect
to
arbitrary
horizontal
transla-
Shinichi
Mochizuki
60
..
..
.
.
⏐
log
⏐
log
⏐
⏐
..
..
.
.
⏐
log
⏐
log
⏐
⏐
Θ
Θ
Θ
Θ
Θ
Θ
Θ
Θ
Θ
.
.
.
−→
•
−→
•
−→
.
.
.
⏐
log
⏐
log
⏐
⏐
.
.
.
−→
•
−→
•
−→
.
.
.
⏐
log
⏐
log
⏐
⏐
•
•
⏐
log
⏐
log
⏐
⏐
Θ
⊇
•
−→
•
⏐
log
⏐
log
⏐
⏐
.
.
.
−→
•
−→
•
−→
.
.
.
⏐
log
⏐
log
⏐
⏐
..
..
.
.
•
•
⏐
log
⏐
log
⏐
⏐
..
..
.
.
Fig.
3.6:
The
entire
log-theta-lattice
and
the
portion
that
is
actually
used
tions,
as
vertically
coric
if
they
are
invariant
with
respect
to
arbitrary
vertical
transla-
tions,
and
as
bi-coric
if
they
are
both
horizontally
and
vertically
coric.
In
this
context,
we
observe
that
—
unlike
any
finite
portion
of
a
vertical
line
of
the
log-theta-lattice!
—
each
[infinite!]
vertical
line
of
the
log-theta-lattice
is
symmetric,
up
to
unique
isomorphism,
with
respect
to
arbitrary
vertical
translations.
As
we
shall
see
in
§3.6,
(iv),
below,
this
is
precisely
why
[cf.
the
portion
of
Fig.
3.6
that
lies
to
the
right
of
the
“⊇”]
it
will
ultimately
be
necessary
to
work
with
the
entire
infinite
vertical
lines
of
the
log-theta-lattice
[i.e.,
as
opposed
to
with
some
finite
portion
of
such
a
vertical
line].
Finally,
we
remark
that
the
two
dimensions
of
the
log-theta-lattice
may
be
thought
of
as
correspond-
ing
to
the
two
underlying
combinatorial
dimensions
of
a
ring
[cf.
the
discussion
of
these
two
dimensions
in
the
case
of
NF’s
and
MLF’s
in
§2.11],
i.e.,
to
addition
and
multiplication.
Indeed,
the
Θ-link
only
involves
the
multiplicative
structure
of
the
rings
that
appear
and,
at
an
extremely
rough
level,
may
be
understood
as
corresponding
to
thinking
of
“numbers”
as
elements
of
the
multiplicative
monoid
of
positive
integers
N
p
N
≥1
∼
=
p
—
where
p
ranges
over
the
prime
numbers,
and
N
denotes
the
additive
monoid
of
nonnegative
integers
—
that
is
to
say,
as
elements
of
an
abstract
monoid
that
admits
automorphisms
that
switch
distinct
prime
numbers
p
1
,
p
2
,
as
well
as
endomorphisms
given
by
raising
to
the
N
-th
power
[cf.
the
discussion
of
§2.4].
By
contrast,
the
log-link
may
be
understood
as
corresponding
to
a
link
between,
or
rotation/juggling
of,
the
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
61
additive
and
multiplicative
structures
at
the
various
completions
of
an
NF
that
is
obtained
by
means
of
the
various
natural
logarithms
defined
on
these
completions
[cf.
the
discussion
of
Example
2.12.3,
(v)].
Here,
we
observe
that
the
noncommutativity
of
the
log-theta-lattice
[which
was
mentioned
at
the
beginning
of
the
present
(ii)]
arises
precisely
from
the
fact
that
the
definition
of
the
Θ-link,
which
only
involves
the
multiplicative
structure
of
the
rings
that
appear,
is
fundamentally
incompatible
with
—
i.e.,
only
makes
sense
once
one
deactivates
—
the
rotation/juggling
of
the
additive
and
multiplicative
structures
that
arises
from
the
log-link.
In
particular,
the
Θ-link
may
only
be
defined
if
one
distinguishes
between
the
domain
and
codomain
of
the
log-link,
i.e.,
between
distinct
vertical
coordinates
in
a
single
vertical
line
of
the
log-theta-lattice.
Moreover,
this
state
of
affairs,
i.e.,
which
requires
one
[in
order
to
define
the
Θ-link!]
to
distinguish
the
ring
structures
in
the
domain
and
codomain
of
the
log-
link
[which
are
related
in
a
non-ring-theoretic
fashion
to
one
another
via
the
log-link!],
makes
it
necessary
to
think
of
the
[possibly
tempered]
arithmetic
fundamental
groups
in
the
domain
and
codomain
of
the
log-link
as
being
related
via
indeterminate
isomorphisms
—
i.e.,
as
discussed
in
§2.10;
Example
2.12.3,
(ii)
[cf.
the
discussion
of
[IUTchIII],
Re-
mark
1.2.2,
(vi),
(a);
[IUTchIII],
Remark
1.2.4,
(i);
[IUTchIV],
Remark
3.6.3,
(i)].
This
situation
may
be
understood
by
means
of
the
analogy
with
the
situation
in
complex
Teichmüller
theory:
one
deforms
one
real
dimension
of
the
complex
structure,
while
holding
the
other
real
dimension
fixed
—
an
operation
that
is
only
meaningful
if
these
two
distinct
real
dimensions
are
not
subject
to
rotations,
i.e.,
to
indeterminacies
with
respect
to
the
action
of
S
1
⊆
C
×
[cf.
[IUTchI],
Remark
3.9.3,
(ii),
(iii),
(iv)].
Put
another
way,
the
portion
of
the
log-theta-lattice
that
is
“actually
used”
[cf.
Fig.
3.6]
exhibits
substantial
structural
similarities
to
the
natural
bijection
∼
C
×
\GL
+
(V
)/C
×
→
[0,
1)
t
0
→
t−1
t+1
0
1
[where
R
t
≥
1]
discussed
in
Example
3.2.1,
(iii),
that
is
to
say:
the
deformation
of
holomorphic
structure
“
0
t
0
1
”
may
be
thought
of
as
corresponding
to
the
single
Θ-link
of
this
portion
of
the
log-theta-lattice,
while
62
Shinichi
Mochizuki
the
“C
×
’s”
on
either
side
of
the
“GL
+
(V
)”
may
be
thought
of
as
corresponding,
respectively,
to
the
vertical
lines
of
log-links
—
i.e.,
rotations
of
the
arith-
metic
holomorphic
structure!
—
on
either
side
of
the
single
Θ-link.
In
the
context
of
this
natural
bijection
discussed
in
Example
3.2.1,
(iii),
it
is
of
interest
to
observe
that
this
double
coset
space
“C
×
\GL
+
(V
)/C
×
”
is
also
reminiscent
of
the
double
coset
spaces
associated
to
groups
of
matrices
over
p-adic
fields
that
arise
in
the
theory
of
Hecke
correspondences.
Alternatively,
relative
to
the
analogy
with
the
two
dimensions
of
an
MLF,
if
the
MLF
under
consideration
is
absolutely
unramified,
i.e.,
isomorphic
to
the
quotient
field
of
a
ring
of
Witt
vectors,
then
one
may
think
of
·
the
log-link
as
corresponding
to
the
Frobenius
morphism
in
positive
char-
acteristic,
i.e.,
to
one
of
the
two
underlying
combinatorial
dimensions
—
namely,
the
slope
zero
dimension
—
of
the
MLF
and
of
·
the
Θ-link
as
corresponding
to
the
mixed
characteristic
extension
struc-
ture
of
a
ring
of
Witt
vectors,
i.e.,
to
the
transition
from
p
n
Z/p
n+1
Z
to
p
n−1
Z/p
n
Z,
which
may
be
thought
of
as
corresponding
to
the
other
of
the
two
underlying
combinatorial
dimensions
—
namely,
the
positive
slope
dimension
—
of
the
MLF.
We
refer
to
[IUTchI],
§I4;
[IUTchIII],
Introduction;
[IUTchIII],
Remark
1.4.1,
(iii);
[IUTchIII],
Remark
3.12.4;
[Pano],
§2,
for
more
details
concerning
the
numerous
analo-
gies
between
inter-universal
Teichmüller
theory
and
various
aspects
of
the
p-adic
the-
ory,
such
as
the
canonical
liftings
that
play
a
central
role
in
the
p-adic
Teichmüller
theory
of
[pOrd],
[pTch],
[pTchIn].
(iii)
The
notion
of
a
Frobenioid:
A
Frobenioid
is
an
abstract
category
whose
abstract
categorical
structure
may
be
thought
of,
roughly
speaking,
as
encoding
the
theory
of
divisors
and
line
bundles
on
various
“coverings”
—
i.e.,
nor-
malizations
in
various
finite
separable
extensions
of
the
function
field
—
of
a
given
normal
integral
scheme.
Here,
the
category
of
such
“coverings”
is
referred
to
as
the
base
category
of
the
Frobenioid.
All
of
the
Frobenioids
that
play
a
[non-negligible]
role
in
inter-universal
Teichmüller
theory
are
model
Frobenioids
[cf.
[FrdI],
Theorem
5.2]
whose
base
cat-
egory
corresponds
to
“some
sort
of”
—
that
is
to
say,
possibly
tempered,
in
the
sense
of
[André],
§4;
[Semi],
Example
3.10
—
arithmetic
fundamental
group
[i.e.,
in
the
non-tempered
case,
the
étale
fundamental
group
of
a
normal
integral
scheme
of
finite
type
over
some
sort
of
“arithmetic
field”].
In
particular,
all
of
the
Frobenioids
that
play
a
[non-negligible]
role
in
inter-universal
Teichmüller
theory
are
essentially
equivalent
to
a
collection
of
data
as
follows
that
satisfies
certain
properties:
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
63
·
a
topological
group,
i.e.,
the
[possibly
tempered]
arithmetic
fundamental
group;
·
for
each
open
subgroup
of
the
topological
group,
an
abelian
group,
called
the
rational
function
monoid,
i.e.,
since
it
is
a
category-theoretic
abstraction
of
the
multiplicative
group
of
rational
functions
on
the
“covering”
corresponding
to
the
given
open
subgroup;
·
for
each
open
subgroup
of
the
topological
group,
an
abelian
monoid,
called
the
divisor
monoid,
i.e.,
since
it
is
a
category-theoretic
abstraction
of
the
monoid
of
Weil
divisors
on
the
“covering”
corresponding
to
the
given
open
subgroup.
In
particular,
such
Frobenioids
may
be
thought
of
as
category-theoretic
abstractions
of
various
aspects
of
the
multiplicative
portion
of
the
ring
structure
of
a
normal
integral
scheme.
We
refer
to
§3.5
below
for
more
remarks
on
the
use
of
Frobenioids
in
inter-
universal
Teichmüller
theory.
(iv)
[Θ
±ell
N
F
-]Hodge
theaters
as
“tautological
solutions”
to
a
purely
combinatorial
problem:
The
Θ
±ell
N
F
-Hodge
theater
associated
to
a
given
col-
lection
of
initial
Θ-data
as
in
(i)
is
a
somewhat
complicated
system
of
Frobenioids
[cf.
[IUTchI],
Definition
6.13,
(i)].
The
topological
group
data
for
these
Frobenioids
arises
from
various
subquotients
of
the
[possibly
tempered]
arithmetic
fundamental
groups
of
the
hyperbolic
orbicurves
discussed
in
(i).
The
rational
function
monoid
data
for
these
Frobenioids
arises
from
the
multiplicative
groups
of
nonzero
elements
of
various
finite
extensions
of
the
number
field
F
mod
of
(i)
or
localizations
[i.e.,
completions]
of
such
NF’s
at
valuations
lying
over
valuations
∈
V.
The
divisor
monoid
data
for
these
Frobenioids
arises,
in
the
case
of
NF’s,
from
the
monoid
of
effective
arithmetic
divisors
[in
the
sense
of
diophantine
geometry
—
cf.,
e.g.,
[GenEll],
§1;
[FrdI],
Example
6.3],
possibly
with
real
coefficients,
and,
in
the
case
of
localizations
of
NF’s,
from
the
nonnegative
portion
of
the
value
group
of
the
associated
valuation,
possibly
tensored
over
Z
with
R.
[In
fact,
at
valuations
in
V
bad
,
an
additional
type
of
Frobenioid,
called
a
tempered
Frobenioid,
also
appears
—
cf.
the
discussion
of
§3.4,
(iv);
§3.5,
below.]
For
instance,
in
the
case
of
localizations
at
valuations
∈
V
good
∩
V
non
,
one
Frobenioid
that
appears
quite
frequently
in
inter-universal
Teichmüller
theory
consists
of
data
that
is
essentially
equivalent
to
the
data
“Π
X
O
k
”
considered
in
Example
2.12.3,
(ii).
[In
the
case
of
valuations
∈
V
bad
,
“Π
X
”
is
replaced
by
the
corresponding
tempered
arithmetic
fundamental
group;
in
the
case
of
valuations
∈
V
arc
(⊆
V
good
),
one
applies
the
theory
of
[AbsTopIII],
§2.]
In
general,
the
Frobenioids
obtained
by
applying
the
operation
of
“passing
to
real
coefficients”
are
referred
to
as
realified
Frobenioids
[cf.
[FrdI],
Proposition
5.3].
The
system
of
Frobenioids
that
64
Shinichi
Mochizuki
constitutes
a
Θ
±ell
N
F
-Hodge
theater
determines,
by
passing
to
the
associated
system
of
base
categories,
an
apparatus
that
is
referred
to
as
a
D-Θ
±ell
N
F
-Hodge
theater
[cf.
[IUTchI],
Definition
6.13,
(ii)].
The
purpose
of
considering
such
systems
of
Frobenioids
lies
in
the
goal
of
reassembling
the
distribution
of
primes
in
the
number
field
K
[cf.
the
discussion
of
[IUTchI],
Remark
4.3.1]
in
such
a
way
as
to
render
possible
the
construction
of
some
sort
of
global
version
of
the
“Gaussian
distribution”
2
{q
j
}
j=1,...,l
discussed
at
the
end
of
§2.4,
i.e.,
which,
a
priori,
is
only
defined
at
the
valuations
∈
V(K)
non
at
which
E
K
has
bad
multiplicative
reduction
[such
as,
for
instance,
the
valuations
∈
V
bad
].
This
“global
version”
amounts
to
the
local
and
global
value
group
portions
of
the
data
that
appears
in
the
domain
portion
of
the
Θ-link
[cf.
(vii)
below].
The
reassembling,
referred
to
above,
of
the
distribution
of
primes
in
the
number
field
K
was
one
of
the
fundamental
motivating
issues
for
the
author
in
the
development
of
the
absolute
mono-
anabelian
geometry
of
[AbsTopIII],
i.e.,
of
a
version
of
anabelian
geometry
that
differs
fundamentally
from
the
well-known
anabelian
result
of
Neukirch-Uchida
concerning
absolute
Galois
groups
of
NF’s
[cf.,
e.g.,
[NSW],
Chapter
XII,
§2]
in
[numerous
ways,
but,
in
particular,
in]
that
its
reconstruction
of
an
NF
does
not
depend
on
the
distribution
of
primes
in
the
NF
[cf.
the
discussion
of
[IUTchI],
Remarks
4.3.1,
4.3.2].
The
problem,
referred
to
above,
of
constructing
a
sort
of
“global
Gaussian
distribution”
may
in
fact
easily
be
seen
to
be
essentially
equivalent
to
the
“purely
combinatorial”
problem
of
constructing
a
“global
multiplicative
subspace”
[cf.
the
discussion
of
§2.3],
together
with
a
“global
canonical
generator”,
i.e.,
more
precisely:
a
one-dimensional
F
l
-
subspace
of
the
two-dimensional
F
l
-vector
space
E
K
[l]
of
l-torsion
points
of
the
elliptic
curve
E
K
,
together
with
a
generator,
well-defined
up
to
multiplication
by
±1,
of
the
quotient
of
E
K
[l]
by
this
one-dimensional
F
l
-subspace,
such
that
this
subspace
and
generator
coincide,
at
the
valuations
∈
V(K)
that
lie
over
valuations
∈
V
bad
mod
,
with
a
certain
canonical
such
subspace
and
generator
that
arise
from
a
generator
[again,
well-defined
up
to
multiplication
by
±1]
of
the
Galois
group
[isomorphic
to
Z]
of
the
well-known
infinite
covering
of
a
Tate
curve.
Here,
we
note
that
such
a
“global
canonical
generator”
determines
a
bijection,
which
is
well-defined
up
to
multiplication
by
±1,
of
the
quotient
referred
to
above
with
the
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
·
−
·
−
·
−
...
−
·
−
·
−
·
·
−
·
−
·
−
...
−
·
−
·
−
·
K
=
V
⊆
V(K)
\
V
Gal(K/F
)
→
GL
2
(F
l
)
...
·
−
·
−
·
−
...
−
·
−
·
−
·
·
−
·
−
·
−
...
−
·
−
·
−
·
65
⊆
V(K)
\
V
⊆
V(K)
\
V
⏐
⏐
F
mod
·
−
·
−
·
−
...
−
·
−
·
−
·
=
V(F
mod
)
Fig.
3.7:
Prime-strips
as
“sections”
of
Spec(K)
→
Spec(F
mod
)
underlying
additive
group
of
F
l
.
In
a
word,
the
combinatorial
structure
of
a
Θ
±ell
N
F
-
Hodge
theater
furnishes
a
sort
of
“tautological
solution”
to
the
purely
combinatorial
problem
referred
to
above
by
simply
ignoring
the
valuations
∈
V(K)\V,
for
instance,
by
working
only
with
Frobenioids
—
i.e.,
in
effect,
arithmetic
divisors/line
bundles
—
that
arise
from
arithmetic
divisors
supported
on
the
set
of
valuations
V
(⊆
V(K)),
i.e.,
as
opposed
to
on
the
entire
set
V(K)
[cf.
the
discussion
of
[IUTchI],
Remark
4.3.1].
Here,
we
note
that
this
sort
of
operation
of
discarding
certain
of
the
primes
of
an
NF
can
only
be
performed
if
one
forgets
the
additive
structure
of
an
NF
[i.e.,
since
a
sum
of
elements
of
an
NF
that
are
invertible
at
a
given
nonempty
set
of
primes
is
no
longer
necessarily
invertible
at
those
primes!
—
cf.
[AbsTopIII],
Remark
5.10.2,
(iv)]
and
works
only
with
multiplicative
structures,
e.g.,
with
Frobenioids.
Collections
of
local
data
—
consisting,
say,
of
local
Frobenioids
or
local
[possibly
tempered]
arithmetic
fundamental
groups
—
indexed
by
the
elements
of
V
are
referred
to
as
prime-strips
[cf.
Fig.
3.7
above;
[IUTchI],
Fig.
I1.2,
and
the
surrounding
discussion].
In
a
word,
prime-strips
may
be
thought
of
as
a
sort
of
monoid-
or
Galois-theoretic
version
of
the
classical
notion
of
adèles/idèles.
(v)
The
symmetries
of
a
Θ
±ell
N
F
-Hodge
theater:
Once
the
“tautological
so-
Shinichi
Mochizuki
66
lution”
furnished
by
the
combinatorial
structure
of
a
Θ
±ell
N
F
-Hodge
theater
is
applied,
the
quotient
of
E
K
[l]
discussed
in
(iv)
corresponds
to
the
quotient
“Q”
of
(i),
i.e.,
in
effect,
to
the
set
of
cusps
of
the
hyperbolic
curve
X
K
of
(i).
One
may
then
consider
additive
and
multiplicative
symmetries
=
F
l
{±1},
F
±
l
def
F
=
F
×
l
/{±1}
l
def
—
where
F
×
l
=
F
l
\
{0},
and
±1
acts
on
F
l
in
the
usual
way
—
on
the
underlying
sets
def
F
l
=
{−l
,
.
.
.
,
−1,
0,
1,
.
.
.
,
l
},
F
l
=
{1,
.
.
.
,
l
}
def
—
where
l
=
(l
−
1)/2;
the
numbers
listed
in
the
above
display
are
to
be
regarded
modulo
l;
we
think
of
F
l
as
the
quotient
Q
[i.e.,
the
set
of
cusps
of
X
K
and
its
localizations
at
valuations
∈
V]
discussed
above
and
of
F
l
as
a
certain
subquotient
of
Q.
Here,
we
remark
that
this
interpretation
of
the
quotient
Q
as
a
set
of
cusps
of
X
K
induces
a
with
the
group
of
“geometric
automorphisms”
natural
outer
isomorphism
of
F
±
l
Aut
K
(X
K
)
[i.e.,
the
group
of
K-automorphisms
of
the
K-scheme
X
K
],
as
well
as
a
natural
iso-
morphism
of
F
l
with
a
certain
quotient
of
the
image
of
the
group
of
“arithmetic
automorphisms”
Aut(C
K
)
→
Gal(K/F
mod
)
[i.e.,
the
group
of
automorphisms
of
the
algebraic
stack
C
K
,
which,
as
is
easily
verified,
maps
injectively
into
the
Galois
group
Gal(K/F
mod
)].
The
combinatorial
structure
of
a
Θ
±ell
N
F
-Hodge
theater
may
then
be
summarized
as
the
system
of
Frobenioids
obtained
by
localizing
and
gluing
together
various
Frobenioids
or
[possibly
tempered]
arithmetic
fundamental
groups
associated
to
X
K
and
C
K
in
the
fashion
prescribed
by
the
combinatorial
recipe
F
l
⊇
F
×
F
±
l
F
l
F
l
l
—
cf.
Fig.
3.8
below.
Here,
we
remark
that,
in
Fig.
3.8:
and
F
·
the
squares
with
actions
by
F
±
l
l
correspond
to
Frobenioids
or
arith-
metic
fundamental
groups
that
arise
from
X
K
and
C
K
,
respectively;
·
each
of
the
elements
of
F
l
or
F
l
that
appears
in
parentheses
“(.
.
.)”
corre-
sponds
to
a
single
prime-strip;
·
each
portion
enclosed
in
brackets
“[.
.
.]”
corresponds
to
a
single
prime-strip;
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
{±1}
−l
<
.
.
.
<
−1
<
0
<
1
<
.
.
.
<
l
⇑
−l
<
.
.
.
<
−1
<
0
<
1
<
.
.
.
<
l
⇒
glue!
⇐
1
<
...
<
l
⇑
1
<
...
<
l
⇓
⇓
±
→
±
→
F
±
l
F
l
↑
↓
±
←
±
.
.
.
cf.
ordinary
monodromy,
additive
symmetries!
67
↑
↓
←
.
.
.
cf.
supersingular
monodromy,
toral
symmetries!
Fig.
3.8:
The
combinatorial
structure
of
a
Θ
±ell
N
F
-Hodge
theater:
a
bookkeeping
apparatus
for
l-torsion
points
·
the
arrows
“⇑”
correspond
to
the
relation
of
passing
from
the
various
indi-
vidual
elements
of
F
l
or
F
l
[i.e.,
one
prime-strip
for
each
individual
element]
to
the
entire
set
F
l
or
F
l
[i.e.,
one
prime-strip
for
the
entire
set];
·
the
arrows
“⇓”
correspond
to
the
relation
of
regarding
the
set
of
elements
in
parentheses
“(.
.
.)”
with
fixed
labels
as
the
underlying
set
of
a
set
equipped
or
F
with
an
action
by
F
±
l
l
;
·
the
gluing
is
the
gluing
prescribed
by
the
surjection
F
l
⊇
F
×
l
F
l
.
In
this
context,
it
is
important
to
keep
in
mind
that
±ell
N
F
-Hodge
theater
play
a
funda-
the
F
±
l
-
and
F
l
-symmetries
of
a
Θ
mental
role
in
the
Kummer-theoretic
aspects
of
inter-universal
Teichmüller
theory
that
are
discussed
in
§3.6
below.
±ell
N
F
-Hodge
theater
may
As
remarked
in
§2.4,
the
F
±
l
-
and
F
l
-symmetries
of
a
Θ
be
thought
of
as
corresponding,
respectively,
to
the
additive
and
multiplicative/toral
symmetries
of
the
classical
upper
half-plane
[cf.
[IUTchI],
Remark
6.12.3,
(iii);
[BogIUT],
±ell
N
F
-Hodge
the-
for
more
details].
Alternatively,
the
F
±
l
-
and
F
l
-symmetries
of
a
Θ
ater
may
be
thought
of
as
corresponding,
respectively,
to
the
[unipotent]
ordinary
and
[toral]
supersingular
monodromy
—
i.e.,
put
another
way,
to
the
well-known
structure
of
the
p-Hecke
correspondence
—
that
occurs
in
the
well-known
classical
68
Shinichi
Mochizuki
p-adic
theory
surrounding
the
moduli
stack
of
elliptic
curves
over
the
p-adic
integers
Z
p
[cf.
the
discussion
of
[IUTchI],
Remark
4.3.1;
[IUTchII],
Remark
4.11.4,
(iii),
(c)].
(vi)
log-links:
Each
vertical
arrow
•
⏐
log
⏐
•
of
the
log-theta-lattice
relates
the
various
copies
of
“Π
X
O
k
”
[cf.
the
discussion
at
the
beginning
of
(iv)]
that
lie
in
the
prime-strips
of
the
domain
Θ
±ell
N
F
-Hodge
theater
“•”
of
the
log-link
to
the
corresponding
copy
of
“Π
X
O
k
”
that
lies
in
a
prime-strip
of
the
codomain
Θ
±ell
N
F
-Hodge
theater
“•”
of
the
log-link
in
the
fashion
prescribed
by
the
arrow
“log”
of
the
diagram
of
Example
2.12.3,
(iii)
[with
suitable
modifications
involving
tempered
arithmetic
fundamental
groups
at
the
valuations
∈
V
bad
or
the
theory
of
[AbsTopIII],
§2,
at
the
valuations
∈
V
arc
(⊆
V
good
)].
In
particular,
the
log-link
may
be
thought
of
as
“lying
over”
an
isomorphism
between
the
respective
copies
of
“Π
X
”
which
is
indeterminate
since
[cf.
the
discussion
of
§2.10;
the
discussion
at
the
end
of
Example
2.12.3,
(ii)]
the
two
copies
of
“Π
X
”
must
be
regarded
as
distinct
abstract
topological
groups.
Put
another
way,
from
the
point
of
view
of
the
discussion
at
the
beginning
of
(iv),
the
log-link
induces
an
indeterminate
isomorphism
between
the
D-Θ
±ell
N
F
-
Hodge
theaters
associated
to
the
Θ
±ell
N
F
-Hodge
theaters
“•”
in
the
domain
and
codomain
of
the
log-link,
that
is
to
say,
these
D-Θ
±ell
N
F
-Hodge
theaters
associated
to
the
•’s
of
the
log-theta-lattice
are
vertically
coric
[cf.
[IUTchIII],
Theorem
1.5,
(i)].
Now
recall
from
Example
2.12.3,
(i),
that
each
abstract
topological
group
“Π
X
”
may
be
regarded
as
the
input
data
for
a
functorial
algorithm
that
allows
one
to
reconstruct
the
base
field
[in
this
case
an
MLF]
of
the
hyperbolic
curve
“X”.
Put
another
way,
from
the
point
of
view
of
the
terminology
discussed
in
§2.7,
(vii),
each
copy
of
“Π
X
”
may
be
regarded
as
an
arithmetic
holomorphic
structure
on
the
quotient
group
“Π
X
G
k
”
associated
to
Π
X
[cf.
the
discussion
of
Example
2.12.3,
(i)].
Indeed,
this
is
precisely
the
point
of
the
analogy
between
the
fundamental
prototypical
examples
—
i.e.,
Examples
3.2.1,
3.2.2
—
of
the
phenomenon
of
multiradiality.
The
various
“X’s”
that
occur
in
a
Θ
±ell
N
F
-Hodge
theater
are
certain
finite
étale
coverings
of
localizations
of
the
hyperbolic
curve
X
K
at
various
valuations
∈
V.
These
finite
étale
coverings
are
X
in
hyperbolic
curves
over
K
v
which
are
denoted
X
v
in
the
case
of
v
∈
V
bad
and
−
→
v
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
69
the
case
of
v
∈
V
good
[cf.
[IUTchI],
Definition
3.1,
(e),
(f)].
On
the
other
hand,
we
recall
from
[AbsTopIII],
Theorem
1.9
[cf.
also
[AbsTopIII],
Remark
1.9.2],
that
this
functorial
algorithm
may
also
be
applied
to
the
hyperbolic
orbicurves
X
K
,
C
K
,
or
C
F
mod
,
i.e.,
whose
base
fields
are
NF’s,
in
a
fashion
that
is
functorial
[cf.
the
discussion
of
[IUTchI],
Remarks
3.1.2,
4.3.2]
with
respect
to
passing
to
finite
étale
coverings,
as
well
as
with
respect
to
localization
at
valuations
of
∈
V
[cf.
also
the
theory
of
[AbsTopIII],
§2,
in
the
case
of
valuations
∈
V
arc
].
That
is
to
say,
in
summary,
the
various
[possibly
tempered,
in
the
case
of
valuations
∈
V
bad
]
arithmetic
fundamental
groups
of
finite
étale
coverings
of
C
F
mod
[such
as
X
K
,
C
K
,
or
C
F
mod
itself]
and
their
localizations
at
valuations
∈
V
that
appear
in
a
Θ
±ell
N
F
-
Hodge
theater
may
be
regarded
as
abstract
representations
of
the
arithmetic
holomorphic
structure
[i.e.,
ring
structure
—
cf.
the
discussion
of
§2.7,
(vii)]
of
the
various
base
fields
of
these
hyperbolic
orbicurves.
Moreover,
this
state
of
affairs
motivates
the
point
of
view
that
the
various
localizations
and
gluings
that
occur
in
the
structure
of
a
single
Θ
±ell
N
F
-Hodge
theater
[cf.
Fig.
3.8]
or,
as
just
described,
in
the
structure
of
a
log-link
[i.e.,
a
vertical
arrow
of
the
log-theta-lattice
—
cf.
Fig.
3.6]
may
be
thought
of
as
arithmetic
analytic
continuations
between
various
NF’s
along
the
various
gluings
of
prime-strips
that
occur
[cf.
the
discussion
of
[IUTchI],
Remarks
4.3.1,
4.3.2,
4.3.3,
5.1.4].
In
this
context,
it
is
of
interest
to
observe
that,
at
a
technical
level,
these
arithmetic
an-
alytic
continuations
are
achieved
by
applying
the
mono-anabelian
theory
of
[AbsTopIII],
§1
[or,
in
the
case
of
archimedean
valuations,
the
theory
of
[AbsTopIII],
§2].
Moreover,
this
mono-anabelian
theory
of
[AbsTopIII],
§1,
is,
in
essence,
an
elementary
consequence
of
the
theory
of
Belyi
cuspidalizations
developed
in
[AbsTopII],
§3
[cf.
[AbsTopIII],
Remark
1.11.3].
Here,
we
recall
that
the
term
cuspidalization
refers
to
a
functorial
algo-
rithm
in
the
arithmetic
fundamental
group
of
a
hyperbolic
curve
for
reconstructing
the
arithmetic
fundamental
group
of
some
dense
open
subscheme
of
the
hyperbolic
curve.
In
particular,
by
considering
Kummer
classes
of
rational
functions
[cf.
the
discussion
of
Example
2.13.1,
(i)],
cuspidalization
may
be
thought
of
as
a
sort
of
“equivalence”
between
the
function
theory
on
a
hyperbolic
curve
and
the
function
theory
on
a
dense
open
subscheme
of
the
hyperbolic
curve
—
a
formulation
that
is
very
formally
remi-
niscent
of
the
classical
notion
of
analytic
continuation.
The
Belyi
cuspidalizations
developed
in
[AbsTopII],
§3,
are
achieved
as
a
formal
consequence
of
the
elementary
observation
that
70
Shinichi
Mochizuki
any
“sufficiently
small”
dense
open
subscheme
U
of
the
hyperbolic
curve
P
given
by
removing
three
points
from
the
projective
line
may
be
regarded
—
via
the
use
of
a
suitable
Belyi
map!
—
as
a
finite
étale
covering
of
P
;
in
particular,
the
arithmetic
fundamental
group
of
U
may
be
recovered
from
the
arithmetic
fundamental
group
of
P
by
considering
a
suitable
open
subgroup
of
the
arithmetic
fundamental
group
of
P
[cf.
[AbsTopII],
Example
3.6;
[AbsTopII],
Corollaries
3.7,
3.8,
for
more
details].
This
state
of
affairs
is
all
the
more
fascinating
in
that
the
well-known
construction
of
Belyi
maps
via
an
induction
on
the
degree
over
Q
of
the
ramification
locus
of
certain
rational
maps
between
two
projective
lines
is
[cf.
the
discussion
of
[IUTchI],
Remark
5.1.4]
highly
reminiscent
of
the
well-known
Schwarz
lemma
of
elementary
complex
analysis,
i.e.,
to
the
effect
that
the
absolute
value,
relative
to
the
respective
Poincaré
metrics,
of
the
derivative
at
any
point
of
a
holomorphic
map
between
copies
of
the
unit
disc
is
≤
1.
(vii)
Θ-links:
Various
versions
of
the
“Θ-link”
are
defined
in
[IUTchI],
[IUTchII],
[IUTchIII]
—
cf.
[IUTchI],
Corollary
3.7,
(i);
[IUTchII],
Corollary
4.10,
(iii);
[IUTchIII],
Definition
3.8,
(ii).
In
the
present
paper,
we
shall
primarily
be
interested
in
the
version
of
[IUTchIII],
Definition
3.8,
(ii).
The
versions
of
[IUTchII],
Corollary
4.10,
(iii),
are
partially
simplified
versions
of
the
version
that
one
is
ultimately
interested
in
[i.e.,
the
version
of
[IUTchIII],
Definition
3.8,
(ii)],
while
the
version
of
[IUTchI],
Corollary
3.7,
(i),
is
an
even
more
drastically
simplified
version
of
these
partially
simplified
versions.
The
Θ-link
may
be
understood,
roughly
speaking,
as
a
realization
of
the
version
of
the
assignment
“q
→
q
N
”
considered
in
the
final
portion
of
the
discussion
of
§2.4,
i.e.,
the
assignment
2
“q
→
{q
j
}
j=1,...,l
”
given
by
taking
a
sort
of
symmetrized
average
as
“N
”
varies
over
the
values
j
2
,
for
j
=
1,
.
.
.
,
l
.
At
a
more
technical
level,
the
Θ-link
Θ
•
−→
•
is
a
gluing
between
two
Θ
±ell
N
F
-Hodge
theaters
“•”,
via
an
indeterminate
[cf.
the
discussion
of
§2.10]
isomorphism
between
certain
gluing
data
arising
from
the
domain
Θ
±ell
N
F
-Hodge
theater
“•”
and
certain
gluing
data
arising
from
the
codomain
Θ
±ell
N
F
-
Hodge
theater
“•”.
The
gluing
data
that
arises
from
the
domain
“•”
of
the
Θ-link
is
as
follows:
(a
Θ
)
[Local]
unit
group
portion:
Consider,
in
the
notation
of
§2.11,
§2.12
[cf.,
es-
pecially,
Example
2.12.3,
(iv)],
the
ind-topological
monoid
equipped
with
an
action
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
71
by
a
topological
group
G
k
O
k
×μ
,
where
we
note
that
O
k
×μ
is
a
Q
p
-vector
space.
H
Observe
that
for
each
open
subgroup
H
⊆
G
k
,
which
determines
a
subfield
k
⊆
k
of
H-invariants
of
k,
the
image
of
O
×
H
in
O
k
×μ
determines
an
integral
structure,
or
k
“lattice”
[i.e.,
a
finite
free
Z
p
-module],
in
the
Q
p
-vector
subspace
of
H-invariants
(O
k
×μ
)
H
of
O
k
×μ
.
[Here,
we
note
that
the
theory
of
the
p-adic
logarithm
determines
H
a
natural
isomorphism
between
this
subspace
and
the
Q
p
-vector
space
k
.]
For
each
v
∈
V
non
,
we
take
the
unit
group
portion
data
at
v,
to
be
this
data
⊆
(O
k
×μ
)
H
}
H
)
(G
k
O
k
×μ
,
{O
×μ
H
k
—
i.e.,
which
we
regard
as
an
ind-topological
monoid
equipped
with
an
action
by
a
topological
group,
together
with,
for
each
open
subgroup
of
the
topological
group,
an
def
def
integral
structure
—
in
the
case
k
=
K
v
.
When
k
=
K
v
,
we
shall
write
G
v
=
G
k
.
An
analogous
construction
may
be
performed
for
v
∈
V
arc
.
(b
Θ
)
Local
value
group
portion:
At
each
v
∈
V
bad
,
we
take
the
local
value
group
portion
data
at
v
to
be
the
formal
monoid
[abstractly
isomorphic
to
the
monoid
N]
generated
by
2
{q
j
}
j=1,...,l
v
—
i.e.,
where
q
∈
O
K
is
a
2l-th
root
of
the
q-parameter
q
v
of
the
elliptic
curve
v
v
E
K
at
v;
the
data
of
the
above
display
is
regarded
as
a
collection
of
elements
of
indexed
by
the
elements
of
F
O
K
l
;
each
element
of
this
collection
is
well-defined
v
up
to
multiplication
by
a
2l-th
root
of
unity.
An
analogous,
though
somewhat
more
formal,
construction
may
be
performed
for
v
∈
V
good
.
(c
Θ
)
Global
value
group
portion:
Observe
that
each
of
the
local
formal
monoids
[say,
for
simplicity,
at
v
∈
V
bad
]
of
(b
Θ
)
may
be
realified.
That
is
to
say,
the
corre-
sponding
realified
monoid
is
simply
the
monoid
of
R
≥0
-multiples
[i.e.,
nonnegative
real
multiples]
of
the
image
of
the
given
monoid
[
∼
=
N]
inside
the
tensor
product
∼
⊗
Z
R
of
the
groupification
[
=
Z]
of
this
given
monoid.
Note,
moreover,
that
the
product
formula
of
elementary
algebraic
number
theory
yields
a
natural
notion
of
“finite
collections
of
elements
of
the
groupifications
[
∼
=
R]
of
these
realified
monoids
at
v
∈
V
whose
sum
=
0”.
This
data,
consisting
of
a
realified
monoid
at
each
v
∈
V,
together
with
a
collection
of
“product
formula
relations”,
determines
a
global
re-
alified
Frobenioid.
We
take
the
global
value
group
portion
data
to
be
this
global
realified
Frobenioid.
The
gluing
data
that
arises
from
the
codomain
of
the
Θ-link
is
as
follows:
Shinichi
Mochizuki
72
(a
q
)
[Local]
unit
group
portion:
For
each
v
∈
V,
we
take
the
unit
group
portion
data
at
v
to
be
the
analogous
data,
i.e.,
this
time
constructed
from
the
codomain
“•”
of
the
Θ-link,
to
the
data
of
(a
Θ
).
(b
q
)
Local
value
group
portion:
At
each
v
∈
V
bad
,
we
take
the
local
value
group
portion
data
at
v
to
be
the
formal
monoid
[abstractly
isomorphic
to
the
monoid
N]
generated
by
q
v
—
i.e.,
where
we
apply
the
notational
conventions
of
(b
Θ
).
An
analogous,
though
somewhat
more
formal,
construction
may
be
performed
for
v
∈
V
good
.
(c
q
)
Global
value
group
portion:
We
take
the
global
value
group
portion
data
to
be
the
global
realified
Frobenioid
[cf.
the
data
of
(c
Θ
)]
determined
by
the
realifications
of
the
local
formal
monoids
of
(b
q
)
at
v
∈
V,
together
with
a
naturally
determined
collection
of
“product
formula
relations”.
In
fact,
the
above
description
is
slightly
inaccurate
in
a
number
of
ways:
for
instance,
in
[IUTchI],
[IUTchII],
[IUTchIII],
the
data
of
(a
Θ
),
(b
Θ
),
(c
Θ
),
(a
q
),
(b
q
),
(c
q
)
are
con-
structed
in
a
somewhat
more
intrinsic
fashion
directly
from
the
various
Frobenioids
[and
other
data]
that
constitute
the
Θ
±ell
N
F
-Hodge
theater
“•”
under
consideration.
This
sort
of
intrinsic
construction
exhibits,
in
a
very
natural
fashion,
the
ind-topological
monoids
“O
k
×μ
”
of
(a
Θ
)
and
(a
q
),
the
local
formal
monoids
of
(b
Θ
)
and
(b
q
),
and
the
global
realified
Frobenioids
of
(c
Θ
)
and
(c
q
)
as
Frobenius-
like
objects.
By
contrast,
the
topological
groups
“G
k
”
of
(a
Θ
)
and
(a
q
)
are
étale-like
objects.
In
this
context,
it
is
useful
to
note
—
cf.
the
discussion
of
the
vertical
coricity
of
D-Θ
±ell
N
F
-Hodge
theaters
in
(vi)
—
that
the
unit
group
portion
data
of
(a
Θ
),
(a
q
)
is
horizontally
coric
[cf.
[IUTchIII],
Theorem
1.5,
(ii)],
while
the
portion
of
this
data
constituted
by
the
topological
group
“G
k
”
is
bi-coric
[cf.
[IUTchIII],
Theorem
1.5,
(iii)].
Indeed,
one
way
to
think
of
Frobenius-like
structures,
in
the
context
of
the
log-theta-
lattice,
is
as
structures
that,
at
least
a
priori,
are
confined
to
—
i.e.,
at
least
a
priori,
are
only
defined
in
—
a
fixed
Θ
±ell
N
F
-Hodge
theater
“•”
of
the
log-theta-lattice.
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
73
Here,
we
note
that
since
[just
as
in
the
case
of
the
log-link
—
cf.
the
discussion
in
the
final
portion
of
(ii)!]
the
Θ-link
is
fundamentally
incompatible
with
the
ring
struc-
tures
in
its
domain
and
codomain,
it
is
necessary
to
think
of
the
bi-coric
topological
group
“G
k
”
as
being
only
well-defined
up
to
some
indeterminate
isomorphism
[cf.
the
discussion
of
§2.10;
[IUTchIII],
Remark
1.4.2,
(i),
(ii);
[IUTchIV],
Remark
3.6.3,
(i)].
Thus,
in
summary,
the
Θ-link
induces
an
isomorphism
of
the
unit
group
portion
data
of
(a
Θ
),
(a
q
),
on
the
one
hand,
and
a
dilation,
by
a
factor
given
by
a
sort
of
symmetrized
average
of
the
j
2
,
for
j
=
1,
.
.
.
,
l
,
of
the
local
and
global
value
group
data
of
(b
Θ
),
(c
Θ
),
(b
q
),
(c
q
),
on
the
other.
The
object,
well-defined
up
to
isomorphism,
of
the
global
realified
Frobenioid
of
(c
Θ
)
determined
by
the
unique
collection
of
generators
of
the
local
formal
monoids
of
(b
Θ
)
at
v
∈
V
bad
will
be
referred
to
as
the
Θ-pilot
object
[cf.
[IUTchI],
Definition
3.8,
(i)].
In
a
similar
vein,
the
object,
well-defined
up
to
isomorphism,
of
the
global
realified
Frobenioid
of
(c
q
)
determined
by
the
unique
collection
of
generators
of
the
local
formal
monoids
of
(b
q
)
at
v
∈
V
bad
will
be
referred
to
as
the
q-pilot
object
[cf.
[IUTchI],
Definition
3.8,
(i)].
The
Θ-
and
q-pilot
objects
play
a
central
role
in
the
main
results
of
inter-universal
Teichmüller
theory,
which
are
the
main
topic
of
§3.7
below.
§
3.4.
Kummer
theory
and
multiradial
decouplings/cyclotomic
rigidity
The
first
main
result
of
inter-universal
Teichmüller
theory
[cf.
§3.7,
(i)]
consists
of
a
multiradial
representation
of
the
Θ-pilot
objects
discussed
in
§3.3,
(vii).
Relative
to
the
general
discussion
of
multiradiality
in
§3.1,
this
multiradiality
may
be
understood
as
being
with
respect
to
the
radial
algorithm
(a
Θ
),
(b
Θ
),
(c
Θ
)
→
(a
Θ
)
that
associates
to
the
gluing
data
in
the
domain
of
the
Θ-link
the
horizontally
coric
unit
group
portion
of
this
data
[cf.
the
discussion
of
§3.3,
(vii)].
The
construction
of
this
multiradial
representation
of
Θ-pilot
objects
consists
of
two
steps.
The
first
step,
which
we
discuss
in
detail
in
the
present
§3.4,
is
the
construction
of
multiradial
cyclotomic
rigidity
and
decoupling
algorithms
for
certain
special
types
of
functions
on
the
hyperbolic
curves
under
consideration.
The
second
step,
which
we
discuss
in
detail
in
§3.6
below,
concerns
the
Galois
evaluation
at
certain
special
points
—
i.e.,
evaluation
via
Galois
sections
of
arithmetic
fundamental
groups
—
of
these
functions
to
obtain
certain
special
values
that
act
on
processions
of
log-shells.
Shinichi
Mochizuki
74
(i)
The
essential
role
of
Kummer
theory:
We
begin
with
the
fundamental
observation
that,
despite
the
fact,
for
∈
{Θ,
q},
the
construction
of
the
data
(a
)
(respectively,
(b
);
(c
))
depends
quite
essentially
on
whether
=
Θ
or
=
q,
the
indeterminate
gluing
isomorphism
that
constitutes
a
Θ-link
exists
—
i.e.,
the
data
(a
Θ
)
(respectively,
(b
Θ
);
(c
Θ
))
is
indeed
isomorphic
to
the
data
(a
q
)
(respectively,
(b
q
);
(c
q
))
—
precisely
as
a
consequence
of
the
fact
that
we
regard
the
[ind-topological]
monoids
that
occur
in
this
data
as
abstract
[ind-topological]
monoids
that
are
not
equipped
with
the
auxiliary
data
of
how
these
[ind-topological]
monoids
“happen
to
be
constructed”.
That
is
to
say,
the
inclusion
of
such
auxiliary
data
would
render
the
corresponding
portions
of
data
in
the
domain
and
codomain
of
the
Θ-link
non-
isomorphic!
Such
abstract
[ind-topological]
monoids
are
a
sort
of
prototypical
example
of
the
notion
of
a
Frobenius-like
structure
[cf.
[IUTchIV],
Example
3.6,
(iii)].
A
similar
observation
applies
to
the
copies
of
“O
k
”
that
occur
in
the
discussion
of
the
log-link
in
§3.3,
(vi).
Thus,
in
summary,
it
is
precisely
by
working
with
Frobenius-like
structures
such
as
abstract
[ind-topological]
monoids
or
abstract
[global
realified]
Frobenioids
that
we
are
able
to
construct
the
non-ring/scheme-theoretic
gluing
isomorphisms
[i.e.,
“non-ring/scheme-theoretic”
in
the
sense
that
they
do
not
arise
from
mor-
phisms
of
rings/schemes!]
of
the
log-
and
Θ-links
of
§3.3,
(vi),
(vii)
[cf.
the
discussion
of
[IUTchII],
Remark
3.6.2,
(ii)].
By
contrast,
étale-like
structures
such
as
the
“G
k
’s”
of
(a
Θ
)
and
(a
q
)
[cf.
§3.3,
(vii)]
or
the
“Π
X
’s”
of
§3.3,
(vi),
will
be
used
to
compute
various
portions
of
the
ring/scheme
theory
on
the
opposite
side
of
a
log-
or
Θ-link
via
the
technique
of
mono-anabelian
transport,
as
discussed
in
§2.7,
§2.9,
i.e.,
by
determining
the
sort
of
indeterminacies
that
one
must
admit
in
order
to
render
the
two
systems
of
Kummer
theories
—
which,
we
recall,
are
applied
in
order
to
relate
Frobenius-like
structures
to
corresponding
étale-like
structures
—
in
the
domain
and
codomain
of
the
log-
or
Θ-link
compatible
with
simultaneous
execution.
Here,
we
recall
that
Kummer
classes
are
obtained,
in
essence,
by
considering
cohomology
classes
that
arise
from
the
action
of
various
Galois
or
arithmetic
fundamental
groups
on
the
various
roots
of
elements
of
an
abstract
monoid
[cf.
Examples
2.6.1,
(iii);
2.12.1,
(i);
2.13.1,
(i)].
Thus,
the
key
step
in
rendering
such
Kummer
classes
independent
of
any
Frobenius-like
structures
lies
in
the
algorithmic
construction
of
a
cyclotomic
rigidity
isomorphism
between
the
group
of
torsion
elements
of
the
abstract
monoid
under
consideration
and
some
sort
of
étale-like
cyclotome,
i.e.,
that
is
constructed
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
75
directly
from
the
Galois
or
arithmetic
fundamental
group
under
consideration
[cf.
the
isomorphism
“λ”
of
Example
2.6.1,
(iii),
(iv);
the
isomorphism
“ρ
μ
k
”
of
Example
2.12.1,
(i),
(ii),
(iv);
the
isomorphism
“λ”
of
Example
2.13.1,
(i),
(ii)].
On
the
other
hand,
let
us
observe,
relative
to
the
multiradiality
mentioned
at
the
beginning
of
the
present
§3.4,
that
the
coric
data
“G
k
O
k
×μ
”
admits
a
[nontrivial!]
natural
action
by
Z
×
O
k
×μ
Z
×
[i.e.,
which
is
G
k
-equivariant
and
compatible
with
the
various
integral
structures
that
appear
in
the
coric
data]
that
lifts
to
a
natural
Z
×
-action
on
O
k
×
[cf.
Example
2.12.2,
(i)].
This
Z
×
-action
induces
a
trivial
action
of
Z
×
on
G
k
[hence
also
on
μ
k
(G
k
)],
but
a
nontrivial
action
of
Z
×
on
μ
k
.
In
particular,
this
Z
×
-action
on
O
k
×
is
manifestly
incompatible
with
the
cyclotomic
rigidity
isomorphism
∼
ρ
μ
k
:
μ
k
→
μ
k
(G
k
)
that
was
functorially
constructed
in
Example
2.12.1,
(ii),
hence,
in
light
of
the
func-
toriality
of
this
construction,
does
not
extend
to
an
action
of
Z
×
on
O
k
.
That
is
to
say,
the
naive
approach
just
discussed
to
cyclotomic
rigidity
isomorphisms
via
the
functorial
construction
of
Example
2.12.1,
(ii),
is
incompatible
with
the
requirement
of
multiradiality,
i.e.,
of
the
existence
of
liftings
of
arbitrary
morphisms
between
collections
of
coric
data.
This
discussion
motivates
the
following
approach,
which
is
fundamental
to
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
[IUTchIII],
Remark
2.2.1,
(iii);
[IUTchIII],
Re-
mark
2.2.2]:
in
order
to
obtain
multiradial
cyclotomic
rigidity
isomorphisms
for
the
local
and
global
value
group
data
(b
Θ
)
and
(c
Θ
),
it
is
necessary
to
somehow
decouple
this
data
(b
Θ
)
and
(c
Θ
)
from
the
unit
group
data
of
(a
Θ
).
This
decoupling
is
achieved
in
inter-universal
Teichmüller
theory
by
working
with
certain
special
types
of
functions,
as
described
in
(ii),
(iii),
(iv),
below.
(ii)
Multiradial
decouplings/cyclotomic
rigidity
for
κ-coric
rational
func-
tions:
The
global
realified
Frobenioids
of
(c
Θ
)
may
be
interpreted
as
“realifications”
of
certain
categories
of
[l
-tuples,
indexed
by
j
=
1,
.
.
.
,
l
,
of
]
arithmetic
line
bundles
on
the
number
field
F
mod
.
The
ring
structure
—
i.e.,
both
the
additive
“”
and
mul-
tiplicative
“”
structures
—
of
copies
of
this
number
field
F
mod
is
applied,
ultimately,
in
inter-universal
Teichmüller
theory,
in
order
to
relate
these
global
realified
Frobe-
nioids
of
(c
Θ
),
which
are,
in
essence,
a
multiplicative
notion,
to
the
interpretation
of
76
Shinichi
Mochizuki
arithmetic
line
bundles
in
terms
of
log-shells,
which
are
modules,
i.e.,
whose
group
law
is
written
additively
[cf.
[IUTchIII],
Remarks
3.6.2,
3.10.1]
—
an
interpretation
with
respect
to
which
global
arithmetic
degrees
correspond
to
log-volumes
of
certain
regions
inside
the
various
log-shells
at
each
v
∈
V
[cf.
the
discussion
of
§2.2].
Thus,
in
summary,
the
essential
structure
of
interest
that
gives
rise
to
the
data
of
(c
Θ
)
consists
of
copies
of
the
number
field
F
mod
indexed
by
j
=
1,
.
.
.
,
l
.
In
particular,
the
Kummer
theory
[cf.
the
discussion
of
(i)!]
concerning
the
data
of
(c
Θ
)
revolves
around
the
Kummer
theory
of
such
copies
of
the
number
field
F
mod
.
As
discussed
at
the
beginning
of
the
present
§3.4,
elements
of
such
copies
of
F
mod
will
be
constructed
as
special
values
at
certain
special
points
of
certain
special
types
of
functions.
Here,
it
is
perhaps
of
interest
to
recall
that
this
approach
to
constructing
elements
of
the
base
field
[in
this
case,
the
number
field
F
mod
]
of
a
hyperbolic
curve
—
i.e.,
by
evaluating
Kummer
classes
of
rational
functions
on
the
hyperbolic
curve
at
certain
special
points
—
is
precisely
the
approach
that
is
in
fact
applied
in
the
mono-anabelian
recon-
struction
algorithms
discussed
in
[AbsTopIII],
§1.
As
discussed
at
the
beginning
of
the
present
§3.4
[cf.
also
the
discussion
of
(i)],
the
first
step
in
the
construction
of
multiradial
representations
of
Θ-pilot
objects
to
be
discussed
in
§3.7,
(i),
consists
of
formulating
the
Kummer
theory
of
suitable
special
types
of
rational
functions
in
such
a
way
that
we
obtain
multiradial
cyclotomic
rigidity
isomorphisms
that
involve
a
decoupling
of
this
Kummer
theory
for
rational
functions
from
the
unit
group
data
of
(a
Θ
).
In
the
present
case,
i.e.,
which
revolves
around
the
construction
of
copies
of
F
mod
,
the
desired
formulation
of
Kummer
theory
is
achieved
by
considering
a
cer-
tain
subset
—
called
the
pseudo-monoid
of
κ-coric
rational
functions
[cf.
[IUTchI],
Remark
3.1.7,
(i),
(ii)]
—
of
the
group
[i.e.,
multiplicative
monoid]
×
”
considered
in
Example
2.13.1,
in
the
case
where
the
hyperbolic
curve
“K
X
“X”
is
taken
to
be
the
hyperbolic
curve
X
K
of
§3.3,
(i)
[cf.
[IUTchI],
Remark
3.1.2,
(ii)].
Recall
the
hyperbolic
orbicurve
C
F
mod
discussed
in
§3.3,
(i).
Write
|C
F
mod
|
for
the
coarse
space
|C
F
mod
|
associated
to
C
F
mod
.
Here,
it
is
useful
to
recall
the
well-known
fact
that
|C
F
mod
|
is
isomorphic
to
the
affine
line
over
L.
We
shall
refer
to
the
points
of
the
compactification
of
|C
F
mod
|
that
arise
from
the
2-torsion
points
of
the
elliptic
curve
E
F
other
than
the
origin
as
strictly
critical.
A
κ-coric
rational
function
is
a
rational
function
on
|C
F
mod
|
that
restricts
to
a
root
of
unity
at
each
strictly
critical
point
of
|C
F
mod
|
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
77
and,
moreover,
satisfies
certain
other
[somewhat
less
essential]
technical
conditions
[cf.
[IUTchI],
Remark
3.1.7,
(i)].
Thus,
a
κ-coric
rational
function
on
|C
F
mod
|
may
also
be
regarded,
by
restriction,
as
a
rational
function
on
X
K
.
Although
the
κ-coric
rational
functions
do
not
form
a
monoid
[i.e.,
the
product
of
two
κ-coric
rational
functions
is
not
necessarily
a
κ-coric
rational
function],
it
nevertheless
holds
that
arbitrary
positive
powers
of
κ-coric
rational
functions
are
κ-coric.
Moreover,
every
root
of
unity
in
F
mod
is
κ-coric;
a
rational
function
on
|C
F
mod
|
is
κ-coric
if
and
only
if
some
positive
power
of
the
rational
function
is
κ-coric.
One
verifies
immediately
that
[despite
the
fact
that
the
κ-coric
rational
functions
do
not
form
a
monoid]
these
elementary
properties
that
are
satisfied
by
κ-coric
rational
functions
are
sufficient
for
conducting
Kummer
theory
with
κ-coric
rational
functions.
Then
[cf.
[IUTchI],
Example
5.1,
(v),
for
more
details]:
·
The
desired
decoupling
of
the
pseudo-monoid
of
κ-coric
rational
functions
from
the
unit
group
data
of
(a
Θ
)
is
achieved
by
means
of
the
condition
that
evaluation
at
any
of
the
strictly
critical
points
—
an
operation
that
may
be
performed
at
the
level
of
étale-like
structures,
i.e.,
by
restricting
Kummer
classes
to
decomposition
groups
of
points
—
yields
a
root
of
unity.
·
The
desired
multiradial
cyclotomic
rigidity
isomorphism
is
achieved
by
means
of
the
technique
discussed
in
Example
2.13.1,
(ii)
—
i.e.,
involving
the
elementary
fact
Z
×
=
{1}
Q
>0
—
which
is
applied
to
the
pseudo-monoid
of
κ-coric
rational
functions,
i.e.,
as
×
.
opposed
to
the
entire
multiplicative
monoid
K
X
As
was
mentioned
in
Example
2.13.1,
(ii),
this
approach
has
the
disadvantage
of
being
incompatible
with
the
profinite
topology
of
the
Galois
or
arithmetic
fundamental
groups
involved
[cf.
the
discussion
of
(iii)
below;
§3.6,
(ii),
below;
[IUTchIII],
Remark
2.3.3,
(vii)].
Also,
we
remark
that
although
this
approach
only
allows
one
to
reconstruct
the
desired
cyclotomic
rigidity
isomorphism
up
to
multiplication
by
±1,
this
will
not
yield
any
problems
since
we
are,
in
fact,
only
interested
in
reconstructing
copies
of
the
×
,
which
is
closed
under
inversion
[cf.
the
discussion
of
entire
multiplicative
monoid
F
mod
[IUTchIII],
Remark
2.3.3,
(vi);
[IUTchIII],
Remark
3.11.4].
(iii)
Naive
approach
to
cyclotomic
rigidity
for
theta
functions:
Fix
v
∈
V
.
Denote
by
means
of
a
subscript
v
the
result
of
base-changing
objects
over
K
to
K
v
.
Thus,
X
v
is
a
“once-punctured
Tate
curve”
over
K
v
,
hence
determines
a
one-
pointed
stable
curve
of
genus
one
X
v
over
the
ring
of
integers
O
K
v
of
K
v
[where,
for
simplicity,
we
omit
the
notation
for
the
single
marked
point,
which
arises
from
the
cusp
of
X
v
].
In
particular,
the
dual
graph
of
the
special
fiber
of
X
v
[i.e.,
more
precisely:
of
X
v
]
is
a
“loop”
[i.e.,
more
precisely,
consists
of
a
single
vertex
and
a
single
edge,
both
ends
bad
Shinichi
Mochizuki
78
labels:
orders
of
zeroes:
poles:
−2
−1
0
1
2
···
—
0
∗
∞
—
0
∗
∞
—
0
∗
∞
—
0
∗
∞
—
0
∗
∞
—
···
1
2
2
1
1
2
1
0
1
1
2
1
2
2
Fig.
3.9:
Labels
of
irreducible
components
and
orders
of
zeroes
at
cusps
“∗”
and
poles
at
irreducible
components
“”
of
the
theta
function
Θ̈
v
on
Ÿ
v
of
which
abut
to
the
single
vertex].
In
particular,
the
universal
covering
[in
the
sense
of
classical
algebraic
topology!]
of
this
dual
graph
determines
[what
is
called]
a
tempered
covering
Y
v
→
X
v
[i.e.,
at
the
level
of
models
over
O
K
v
,
a
tempered
covering
Y
v
→
X
v
def
—
cf.
[André],
§4;
[Semi],
Example
3.10],
whose
Galois
group
Z
=
Gal(Y
v
/X
v
)
is
noncanonically
isomorphic
to
Z.
Thus,
the
special
fiber
of
Y
v
[i.e.,
more
precisely:
of
Y
v
]
consists
of
an
infinite
chain
of
copies
of
the
“once-punctured/one-pointed
projective
line”,
in
which
the
“punctures/cusps”
correspond
to
the
points
“1”
of
the
copies
of
the
projective
line,
and
the
point
“∞”
of
each
such
copy
is
glued
to
the
point
“0”
of
the
adjacent
copy
[cf.
the
upper
portion
of
Fig.
3.9
above;
the
discussion
at
the
beginning
of
[EtTh],
§1;
[IUTchII],
Proposition
2.1;
[IUTchII],
Remark
2.1.1].
If
one
fixes
one
such
copy
of
the
once-punctured
projective
line,
together
with
an
iso-
∼
morphism
Z
→
Z,
then
the
natural
action
of
Z
on
Y
v
determines
a
natural
bijection
of
the
set
of
irreducible
components
of
the
special
fiber
of
Y
v
—
or,
alternatively,
of
the
set
of
cusps
of
Y
v
—
with
Z
[cf.
the
“labels”
of
Fig.
3.9].
The
“multiplication
by
2”
endomorphism
of
the
elliptic
curve
E
v
[which
may
be
thought
of
as
the
compactification
of
the
affine
hyperbolic
curve
X
v
]
determines,
via
base-change
by
Y
v
→
X
v
,
a
double
covering
Ÿ
v
→
Y
v
[i.e.,
at
the
level
of
models
over
O
K
v
,
a
double
covering
Ÿ
v
→
Y
v
].
One
verifies
immediately
that
the
set
of
irreducible
components
of
the
special
fiber
of
Ÿ
v
[i.e.,
more
precisely:
of
Ÿ
v
]
maps
bijectively
to
the
set
of
irreducible
components
of
the
special
fiber
of
Y
v
,
while
there
exist
precisely
two
cusps
of
Ÿ
v
over
each
cusp
of
Y
v
.
The
formal
completion
of
Y
v
along
the
smooth
locus
[i.e.,
the
complement
of
“0”
and
“∞”]
of
the
irreducible
component
of
the
special
fiber
labeled
0
is
naturally
isomorphic
∼
[in
a
fashion
compatible
with
a
choice
of
isomorphism
Z
→
Z]
to
a
once-punctured
copy
of
the
multiplicative
group
“G
m
”.
In
particular,
it
makes
sense
to
speak
of
the
standard
multiplicative
coordinate
“U
v
”
on
this
formal
completion,
as
well
as
a
square
root
[well-defined
up
to
multiplication
by
±1]
“
Ü
v
”
of
U
v
on
the
base-change
of
this
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
79
formal
completion
by
Ÿ
v
→
Y
v
.
The
theta
function
def
−
1
Θ̈
v
=
Θ̈
v
(
Ü
v
)
=
q
v
8
·
1
(−1)
n
·
q
v
2
(n+
12
)
2
·
Ü
v
2n+1
n∈Z
may
be
thought
of
as
a
meromorphic
function
on
Ÿ
v
[cf.
[EtTh],
Proposition
1.4,
and
the
preceding
discussion],
whose
zeroes
are
precisely
the
cusps,
with
multiplicity
1,
and
whose
poles
are
supported
on
the
special
fiber
of
Ÿ
v
,
with
multiplicity
[relative
to
a
1
square
root
q
v
2
of
q
v
]
equal
to
j
2
at
the
irreducible
component
labeled
j
[cf.
Fig.
3.9].
In
fact,
in
inter-universal
Teichmüller
theory,
we
shall
mainly
be
interested
in
[a
certain
constant
multiple
of]
the
reciprocal
of
an
l-th
root
of
this
theta
function,
namely,
Θ
v
def
=
√
−1
·
m∈Z
1
q
v
2
(m+
12
)
2
−1
·
1
(−1)
n
·
q
v
2
(n+
12
)
2
n+
12
·
U
v
−
1
l
n∈Z
def
—
which
may
be
thought
of
as
a
meromorphic
function
on
Ÿ
v
=
Ÿ
v
×
X
v
X
v
[cf.
the
notation
of
§3.3,
(vi)]
that
is
normalized
by
the
condition
that
it
assumes
a
value
∈
μ
2l
[i.e.,
a
2l-th
root
of
unity]
at
the
points
of
Ÿ
v
that
lie
over
the
torsion
points
of
E
v
of
order
precisely
4
and,
moreover,
meet
the
smooth
locus
of
the
irreducible
component
def
of
the
special
fiber
of
Ÿ
v
[i.e.,
more
precisely:
of
Ÿ
=
Ÿ
v
×
X
v
X
v
,
where
X
v
and
v
X
v
denote
the
respective
normalizations
of
X
v
in
X
v
and
X
v
]
labeled
0
[i.e.,
the
unique
irreducible
component
of
the
special
fiber
of
Ÿ
v
that
maps
to
the
irreducible
component
of
the
special
fiber
of
Ÿ
v
labeled
0].
Such
points
of
Ÿ
v
are
referred
to
as
zero-labeled
evaluation
points
[cf.,
e.g.,
[IUTchII],
Corollary
2.6].
At
a
very
rough
level,
the
approach
to
multiradial
decouplings/cyclotomic
rigidity
taken
in
the
case
of
the
Kummer
theory
of
special
functions
that
surrounds
the
formal
monoid
of
(b
Θ
)
may
be
understood
as
being
“roughly
similar”
to
the
approach
discussed
in
(ii)
in
the
case
of
the
global
realified
Frobenioids
of
(c
Θ
),
except
that
“κ-coric
rational
functions”
are
replaced
by
[normalized
reciprocals
of
l-th
roots
of]
theta
functions.
[cf.
the
discussion
of
[IUTchII],
Remark
1.1.1,
(v);
[IUTchIII],
Remark
2.3.3].
That
is
to
say,
·
The
desired
decoupling
[which
is
referred
to
in
[EtTh],
as
“constant
multiple
rigidity”]
of
the
[reciprocal
of
an
l-th
root
of
the]
theta
function
from
the
unit
group
data
of
(a
Θ
)
is
achieved
by
means
of
the
condition
that
evaluation
at
any
of
the
zero-labeled
evaluation
points
—
an
operation
that
may
be
performed
at
the
level
of
étale-like
structures,
i.e.,
by
restricting
Kummer
classes
to
decomposition
groups
of
points
—
yields
a
2l-th
root
of
unity.
80
Shinichi
Mochizuki
·
The
desired
multiradial
cyclotomic
rigidity
isomorphism
is,
roughly
speaking,
achieved
by
means
of
the
“mod
N
Kummer
class
version”,
for
various
positive
integers
N
,
of
the
technique
discussed
in
Example
2.13.1,
(ii)
[cf.,
especially,
the
final
display
of
Example
2.13.1,
(ii)]:
that
is
to
say,
such
a
“mod
N
version”
is
possible
–
without
any
{±1}
indeterminacies!
—
precisely
as
a
consequence
of
the
fact
that
the
order
of
each
zero
of
Θ̈
v
at
each
cusp
of
Ÿ
v
is
precisely
one
[cf.,
e.g.,
the
discussion
of
[IUTchIII],
Remark
2.3.3,
(vi)].
In
this
context,
we
remark
that
the
decomposition
groups
of
zero-labeled
evaluation
points
may
be
reconstructed
by
applying
the
theory
of
elliptic
cuspidalizations
de-
veloped
in
[AbsTopII],
§3.
This
theory
proceeds
in
an
essentially
parallel
fashion
to
the
theory
of
Belyi
cuspidalizations
[cf.
the
discussion
of
§3.3,
(vi)].
That
is
to
say,
elliptic
cuspidalizations
are
achieved
as
a
formal
consequence
of
the
elementary
observation
that
the
dense
open
subscheme
of
a
once-punctured
elliptic
curve
obtained
by
removing
the
N
-torsion
points,
for
N
a
positive
integer,
may
be
regarded
—
via
the
use
of
the
“multiplication
by
N
”
endomorphism
of
the
elliptic
curve!
—
as
a
finite
étale
covering
of
the
given
once-punctured
elliptic
curve;
in
particular,
the
arithmetic
fundamental
group
of
such
a
dense
open
subscheme
may
be
recovered
from
the
arithmetic
fundamental
group
of
the
given
once-punctured
elliptic
curve
by
considering
a
suitable
open
subgroup
of
the
latter
arithmetic
fundamental
group
[cf.
[AbsTopII],
Example
3.2;
[AbsTopII],
Corollaries
3.3,
3.4,
for
more
details].
Another
important
observation
in
this
context
[cf.
also
§3.6,
(ii),
below;
[IUTchIII],
Remark
2.3.3,
(vii)]
is
that
the
approach
to
cyclotomic
rigidity
described
above
involv-
ing
“mod
N
Kummer
classes”
is
manifestly
compatible
with
the
topology
of
the
Galois
or
tempered
arithmetic
fundamental
groups
involved.
Since
this
“mod
N
ap-
proach”
depends,
in
an
essential
way,
on
the
fact
that
the
order
of
the
zero
of
Θ̈
v
at
each
cusp
of
Ÿ
v
is
precisely
one
[cf.,
the
discussion
of
[IUTchIII],
Remark
2.3.3,
(vi)],
it
also
serves
to
elucidate
the
importance
of
working
with
the
first
power
of
[reciprocals
of
l-th
roots
of
the]
theta
function,
i.e.,
as
opposed
to
the
M
-th
power,
for
M
≥
2
[cf.
[IUTchII],
Remark
3.6.4,
(iii),
(iv);
[IUTchIII],
Remark
2.1.1,
(iv);
[IUTchIII],
Remark
2.3.3,
(vii)].
On
the
other
hand,
the
approach
described
thus
far
in
the
present
(iii)
has
one
fundamental
deficiency,
namely,
the
fact
that
the
orders
of
the
poles
of
Θ̈
v
are
not
compatible/symmetric
with
respect
to
the
action
of
Z
on
Y
v
[cf.
Fig.
3.9]
implies
that
the
approach
described
thus
far
in
the
present
(iii)
to
multiradial
cyclotomic
rigidity
—
i.e.,
involving,
in
effect,
the
mod
N
Kummer
classes
of
the
[reciprocal
of
an
l-th
root
of
the]
theta
function
—
is
not
compatible
with
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
81
the
Z-symmetries
of
Y
v
[cf.
the
discussion
of
[IUTchII],
Remark
1.1.1,
(v);
[IUTchIII],
Remark
2.3.3,
(iv)].
∼
Here,
we
note
that
the
quotient
Z
→
Z
Z/l
·
Z
=
F
l
may
be
identified
with
the
in
the
discussion
of
symmetries
of
Θ
±ell
N
F
-Hodge
theaters
in
§3.3,
subgroup
F
l
⊆
F
±
l
(v).
Also,
we
remark
that
the
mod
N
Kummer
class
of
the
reciprocal
of
an
l-th
root
of
the
theta
function
is
indeed
compatible
with
the
N
·
l
·
Z-symmetries
of
Y
v
.
This
means
that,
as
one
varies
N
,
the
obstruction
to
finding
a
coherent
system
of
basepoints
—
i.e.,
a
coherent
notion
of
the
“zero
label”
—
of
the
resulting
projective
system
lies
in
∼
∼
R
1
lim
N
·
l
·
Z
→
R
1
lim
N
·
l
·
Z
→
l
·
Z/l
·
Z
=
0
←−
←−
N
N
[cf.
the
discussion
of
[EtTh],
Remark
2.16.1].
Put
another
way,
one
may
only
construct
a
coherent
system
of
basepoints
if
one
is
willing
to
replace
Z
by
its
profinite
comple-
tion,
i.e.,
to
sacrifice
the
discrete
nature
of
Z
(
∼
=
Z).
It
is
for
this
reason
that
the
property
of
compatibility
with,
say,
the
l
·
Z-symmetries
of
Y
v
is
referred
to,
in
[EtTh],
as
discrete
rigidity.
This
sort
of
discrete
rigidity
plays
an
important
role
in
inter-
universal
Teichmüller
theory
since
a
failure
of
discrete
rigidity
would
obligate
one
to
work
with
Z-multiples/powers
of
divisors,
line
bundles,
or
meromorphic
functions
—
a
state
of
affairs
that
would,
for
instance,
obligate
one
to
sacrifice
the
crucial
notion
of
positivity/ampleness
in
discussions
of
divisors
and
line
bundles
[cf.
the
discussion
of
[IUTchIII],
Remark
2.1.1,
(v)].
(iv)
Cyclotomic,
discrete,
and
constant
multiple
rigidity
for
mono-theta
environments:
The
incompatibility
of
the
approach
discussed
in
(iii)
with
Z-symme-
tries
[cf.
the
discussion
at
the
end
of
(iii)]
is
remedied
in
the
theory
of
[EtTh]
by
working
with
mono-theta
environments,
as
follows:
Write
L
X
v
for
the
ample
line
bundle
of
degree
1
on
X
v
determined
by
the
unique
marked
point
of
X
v
and
L
v
L
X
v
def
=
def
=
L
X
v
|
X
v
,
L
X
v
|
Y
v
,
L
v
L̈
v
def
=
def
=
L
v
|
Y
v
,
L
X
v
|
Ÿ
v
,
L̈
v
def
=
L̈
v
|
Ÿ
v
for
the
various
pull-backs,
or
restrictions,
of
L
X
v
.
Then
the
theta
function
Θ̈
v
on
Ÿ
v
may
be
thought
of
as
a
ratio
of
two
sections
of
the
line
bundle
L̈
v
over
Ÿ
v
,
which
may
be
described
as
follows:
·
The
algebraic
section
of
L̈
v
is
the
section
[well-defined
up
to
a
K
v
×
-multiple]
whose
zero
locus
coincides
with
the
locus
of
zeroes
of
Θ̈
v
.
The
pair
consisting
of
the
line
bundle
L
v
and
this
algebraic
section
admits
Z-symmetries
[cf.
Fig.
3.9],
i.e.,
automorphisms
that
lie
over
the
automorphisms
of
Z
=
Gal(Y
v
/X
v
).
·
The
theta
section
of
L̈
v
is
the
section
[well-defined
up
to
a
K
v
×
-multiple]
Shinichi
Mochizuki
82
whose
zero
locus
coincides
with
the
locus
of
poles
of
Θ̈
v
.
One
verifies
im-
mediately
[cf.
Fig.
3.9]
that
the
theta
section
is
not
compatible
with
the
Z-symmetries
of
the
algebraic
section.
The
analogous
operation,
for
this
line
bundle-theoretic
data,
to
considering
various
Kum-
mer
classes
of
the
theta
function
is
the
operation
of
passing
to
the
tempered
arith-
metic
fundamental
group
of
the
G
m
-torsor
L
×
v
associated
to
L
v
or
to
the
morphisms
on
tempered
arithmetic
fundamental
groups
induced
by
the
algebraic
and
theta
sections.
Here,
“mod
N
Kummer
classes”
correspond
to
considering
the
quotient
of
the
tempered
arithmetic
fundamental
group
of
L
×
v
that
corresponds
to
coverings
whose
restriction
to
the
“G
m
fibers”
of
the
G
m
-torsor
L
×
v
is
dominated
by
the
covering
G
m
→
G
m
given
by
raising
to
N
-th
power.
Note
that
neither
the
ratio
of
the
algebraic
and
theta
sections
—
i.e.,
the
theta
func-
tion!
—
nor
the
pair
consisting
of
the
algebraic
and
theta
sections
is
com-
patible
with
the
Z-symmetries
of
Y
v
.
On
the
other
hand,
it
is
not
difficult
to
verify
that
the
following
triple
of
data
is
indeed
compatible,
up
to
isomorphism,
with
the
Z-symmetries
of
Y
v
:
(a
μ−Θ
)
the
G
m
-torsor
L
×
v
;
×
(b
μ−Θ
)
the
group
of
automorphisms
of
L
×
v
generated
by
the
Z-symmetries
of
L
v
and
the
automorphisms
determined
by
multiplication
by
a
constant
∈
K
v
×
;
def
×
(c
μ−Θ
)
the
theta
section
of
L̈
×
v
=
L
v
|
Ÿ
v
.
Indeed,
the
asserted
compatibility
with
the
Z-symmetries
of
Y
v
is
immediate
for
(a
μ−Θ
)
and
(b
μ−Θ
).
On
the
other
hand,
with
regard
to
(c
μ−Θ
),
a
direct
calculation
shows
that
application
of
a
Z-symmetry
has
the
effect
of
multiplying
the
theta
section
by
some
1
meromorphic
function
which
is
a
product
of
integer
powers
of
Ü
v
and
q
v
2
;
moreover,
a
direct
calculation
shows
that
the
group
of
automorphisms
of
(b
μ−Θ
)
is
stabilized
by
con-
jugation
by
the
operation
of
multiplying
by
such
a
meromorphic
function.
That
is
to
say,
by
applying
such
multiplication
operations,
we
conclude
that
the
triple
of
data
(a
μ−Θ
),
(b
μ−Θ
),
(c
μ−Θ
)
is
indeed
compatible,
up
to
isomorphism,
with
the
Z-symmetries
of
Y
v
,
as
desired
[cf.
[EtTh],
Proposition
2.14,
(ii),
(iii),
for
more
details].
This
argument
motivates
the
following
definition
[cf.
the
discussion
of
[IUTchIII],
Remark
2.3.4]:
The
[mod
N
]
mono-theta
environment
is
defined
by
considering
the
[mod
N
]
tempered
arithmetic
fundamental
group
versions
of
the
“l-th
roots”
of
the
triple
of
data
(a
μ−Θ
),
(b
μ−Θ
),
(c
μ−Θ
)
discussed
above
[cf.
[EtTh],
Definition
2.13,
(ii)].
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
83
[Indeed,
the
data
(a
μ−Θ
),
(b
μ−Θ
),
(c
μ−Θ
)
correspond,
respectively,
to
the
data
of
[EtTh],
Definition
2.13,
(ii),
(a),
(b),
(c).]
In
particular,
the
functoriality
of
the
tempered
arithmetic
fundamental
group
[essentially
—
cf.
[EtTh],
Proposition
2.14,
(ii),
(iii),
for
more
details]
implies
that
a
mono-theta
environment
admits
l
·
Z-symmetries
of
the
desired
type,
hence,
in
particular,
that
it
satisfies
the
crucial
property
of
discrete
rigidity
discussed
in
the
final
portion
of
(iii).
Moreover,
by
forming
suitable
commu-
tators
in
the
group
of
automorphisms
of
L
×
v
,
one
may
recover
the
desired
cyclotomic
rigidity
isomorphism
[cf.
[EtTh],
Corollary
2.19,
(i);
[IUTchII],
Remark
1.1.1,
for
more
details],
i.e.,
that
was
discussed
in
a
“rough
form”
in
(iii),
in
a
fashion
that
is
·
decoupled
from
the
unit
group
data
of
(a
Θ
),
·
manifestly
compatible
with
the
topology
of
the
tempered
arithmetic
fun-
damental
groups
involved
[since
one
works
with
“mod
N
”
mono-theta
environ-
ments!],
and
±ell
N
F
-Hodge
theaters
[cf.
§3.3,
·
compatible
with
the
F
±
l
-symmetries
of
Θ
(v);
[IUTchII],
Remark
1.1.1,
(iv),
(v)].
Indeed,
these
multiradial
decoupling/cyclotomic
rigidity
properties
of
mono-theta
environments
are
the
main
topic
of
[IUTchII],
§1,
and
are
summarized
in
[IUTchII],
Corollaries
1.10,
1.12.
Moreover,
mono-theta
environments
have
both
étale-like
and
Frobenius-like
versions,
i.e.,
they
may
be
constructed
naturally
[cf.
[IUTchII],
Propo-
sition
1.2,
(i),
(ii)]
either
·
from
the
tempered
arithmetic
fundamental
group
[regarded
as
an
ab-
stract
topological
group!]
of
X
v
,
or
·
from
a
certain
“tempered
Frobenioid”,
i.e.,
a
model
Frobenioid
[cf.
§3.3,
(iii)]
obtained
by
considering
suitable
divisors,
line
bundles,
and
meromorphic
functions
on
the
various
tempered
coverings
of
X
v
.
Finally,
we
close
with
the
important
observation
that
the
various
rigidity
properties
of
mono-theta
environments
discussed
above
may
be
regarded
as
essentially
formal
consequences
of
the
quadratic
structure
of
the
commu-
tators
of
the
theta
groups
—
or,
equivalently,
of
the
curvature,
or
first
Chern
class
—
associated
to
the
line
bundle
L
X
v
[cf.
the
discussion
of
[IUTchII],
Remark
1.1.1,
(iv),
(v);
[IUTchIII],
Remark
2.1.1].
This
observation
is
of
interest
in
that
it
shows
that
the
theory
of
[EtTh]
[or,
indeed,
a
sub-
stantial
portion
of
inter-universal
Teichmüller
theory!]
yields
an
interesting
alternative
interpretation
for
the
structure
of
theta
groups
to
the
classical
representation-
theoretic
interpretation,
i.e.,
involving
irreducible
representations
of
theta
groups
[cf.
[IUTchIII],
Remark
2.3.4,
(iv)].
Shinichi
Mochizuki
84
Approach
to
cyclotomic
rigidity
Applied
to
Kummer
theory
surrounding
Uni-/multi-
radiality
Compatibility
with
profinite/
tempered
topologies
Brauer
groups/
local
class
field
theory
for
“G
k
O
k
”
(a
Θ
)
uniradial
compatible
mono-theta
environ-
ments
(b
Θ
)
multiradial
compatible
(c
Θ
)
multiradial
incompatible
κ-coric
rational
functions,
via
×
Z
Q
>0
={1}
Fig.
3.10:
Three
approaches
to
cyclotomic
rigidity
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
85
(v)
Various
approaches
to
cyclotomic
rigidity:
The
discussion
in
the
present
§3.4
of
various
properties
of
the
three
approaches
to
cyclotomic
rigidity
that
appear
in
inter-universal
Teichmüller
theory
is
summarized
in
Fig.
3.10
above.
The
most
naive
ap-
proach,
involving
well-known
properties
from
local
class
field
theory
applied
to
the
data
“G
k
O
k
”
[cf.
Example
2.12.1,
(ii),
(iii),
(iv)],
is
compatible
with
the
profinite
topology
of
the
Galois
or
arithmetic
fundamental
groups
involved,
but
suffers
from
the
fundamental
defect
of
being
uniradial,
i.e.,
of
being
“un-decouplable”
from
the
unit
group
data
of
(a
Θ
)
[cf.
the
discussion
of
(i)].
By
contrast,
the
approaches
discussed
in
(ii)
and
(iv)
involving
κ-coric
rational
functions
and
mono-theta
environments
satisfy
the
crucial
requirement
of
multiradiality,
i.e.,
of
being
“decouplable”
from
the
unit
group
data
of
(a
Θ
).
The
approach
via
mono-theta
environments
also
satisfies
the
important
property
of
being
compatible
with
the
topology
of
the
tempered
arithmetic
fundamental
groups
involved.
By
contrast,
the
approach
via
κ-coric
rational
functions
is
not
compatible
with
the
profinite
topology
of
the
Galois
or
arithmetic
fundamental
groups
involved.
This
incompatibility
in
the
case
of
the
Kummer
theory
surrounding
the
global
data
of
(c
Θ
)
will
not,
however,
pose
a
problem,
since
compatibility
with
the
topologies
of
the
various
Galois
or
[possibly
tempered]
arithmetic
fundamental
groups
involved
will
only
be
of
interest
in
the
case
of
the
Kummer
theory
surrounding
the
local
data
of
(a
Θ
)
and
(b
Θ
)
[cf.
§3.6,
(ii),
below;
[IUTchIII],
Remark
2.3.3,
(vii),
(viii)].
§
3.5.
Remarks
on
the
use
of
Frobenioids
The
theory
of
Frobenioids
was
developed
in
[FrdI],
[FrdII]
as
a
solution
to
the
problem
of
providing
a
unified,
intrinsic
category-theoretic
characterization
of
various
types
of
categories
of
line
bundles
and
divisors
that
frequently
appeared
in
the
author’s
research
on
the
arithmetic
of
hyperbolic
curves
and,
moreover,
seemed,
at
least
from
a
heuristic
point
of
view,
to
be
remarkably
similar
in
structure.
These
papers
[FrdI],
[FrdII]
on
Frobenioids
were
written
in
the
spring
of
2005,
when
the
author
only
had
a
relatively
rough,
sketchy
idea
of
how
to
formulate
inter-universal
Teichmüller
theory.
In
particular,
anyone
who
reads
these
papers
[FrdI],
[FrdII]
—
or
indeed,
the
Frobenioid-
theoretic
portion
of
[EtTh]
—
under
the
expectation
that
they
were
written
as
an
optimally
efficient
presentation
of
precisely
those
portions
of
the
theory
of
Frobenioids
that
are
actually
used
in
inter-universal
Teichmüller
theory
will
undoubtedly
be
disappointed.
In
light
of
this
state
of
affairs,
it
seems
appropriate
to
pause
at
this
point
to
make
a
few
remarks
on
the
use
of
Frobenioids
in
inter-universal
Teichmüller
theory.
First
of
all,
Shinichi
Mochizuki
86
·
at
v
∈
V
arc
,
the
[“archimedean”]
Frobenioids
that
appear
in
inter-universal
Teichmüller
theory
[cf.
[IUTchI],
Example
3.4]
are
essentially
equivalent
to
the
topological
monoid
O
C
[i.e.,
the
multiplicative
topological
monoid
of
nonzero
complex
numbers
of
norm
≤
1]
and
hence
may
be
ignored.
On
the
other
hand,
·
at
v
∈
V
non
,
all
of
the
[“nonarchimedean”]
Frobenioids
that
appear
in
inter-universal
Teichmüller
theory
[cf.,
e.g.,
[IUTchI],
Fig.
I1.2]
—
except
for
the
tempered
Frobenioids
mentioned
in
§3.4,
(iv)!
—
are
essentially
equivalent
to
either
the
data
[consisting
of
an
abstract
ind-topological
monoid
equipped
with
a
continuous
action
by
an
abstract
topological
group]
“G
k
O
k
”
of
Example
2.12.1,
(i),
or
the
data
[consisting
of
an
abstract
ind-topological
monoid
equipped
with
a
continuous
action
by
an
abstract
topological
group]
“Π
X
O
k
”
of
Example
2.12.3,
(ii)
[where
Π
X
is
possibly
replaced
by
the
tempered
arith-
metic
fundamental
group
of
X],
or
the
data
obtained
from
one
of
these
two
types
of
data
by
replacing
“O
k
”
by
some
subquotient
of
“O
k
”
[as
in
Example
2.12.2,
(i),
(ii)].
Moreover,
all
of
these
“nonarchimedean”
Frobenioids
are
model
Frobenioids
[cf.
the
discussion
of
§3.3,
(iii)].
The
only
other
types
of
Frobenioids
—
all
of
which
are
model
Frobenioids
[cf.
the
discussion
of
§3.3,
(iii)]
—
that
appear
in
inter-universal
Te-
ichmüller
theory
are
·
the
[possibly
realified]
global
Frobenioids
associated
to
NF’s
[cf.
the
dis-
cussion
of
§3.4,
(ii)],
which
admit
a
simple
elementary
description
as
categories
of
arithmetic
line
bundles
on
NF’s
[cf.
[FrdI],
Example
6.3;
[IUTchIII],
Example
3.6;
[Fsk],
§2.10,
(i),
(ii)];
·
the
tempered
Frobenioids
mentioned
in
§3.4,
(iv).
Here,
we
note
that
these
last
two
examples
—
i.e.,
global
Frobenioids
and
tempered
Frobenioids
—
differ
fundamentally
from
the
previous
examples,
which
were
essentially
equiva-
lent
to
an
ind-topological
monoid
that
was,
in
some
cases,
equipped
with
a
con-
tinuous
action
by
a
topological
group,
in
that
their
Picard
groups
[cf.
[FrdI],
Theorem
5.1]
admit
non-torsion
elements.
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
87
Indeed,
global
Frobenioids
contain
objects
corresponding
to
arithmetic
line
bundles
whose
arithmetic
degree
is
=
0,
while
tempered
Frobenioids
contain
objects
correspond-
ing
to
line
bundles
for
which
arbitrary
positive
tensor
powers
are
nontrivial
such
as
[strictly
speaking,
the
pull-back
to
Ÿ
v
of]
the
line
bundle
“L
v
”
of
§3.4,
(iv).
Finally,
we
remark
that
although
the
theory
of
tempered
Frobenioids,
which
is
developed
in
[EtTh],
§3,
§4,
§5,
is
somewhat
complicated,
the
only
portions
of
these
tempered
Frobenioids
that
are
actually
used
in
inter-universal
Teichmüller
theory
are
the
portions
discussed
in
§3.4,
(iii),
(iv),
i.e.,
(a
t−F
)
the
“theta
monoids”
generated
by
local
units
[i.e.,
“O
×
”]
and
nonnegative
powers
of
roots
of
the
[reciprocals
of
l-th
roots
of]
theta
functions
that
are
con-
structed
from
tempered
Frobenioids
[cf.
[IUTchI],
Example
3.2;
[IUTchII],
Example
3.2,
(i)];
(b
t−F
)
the
mono-theta
environments
constructed
from
tempered
Frobenioids
[cf.
[IUTchII],
Proposition
1.2,
(ii)],
which
are
related
to
the
monoids
of
(a
t−F
),
in
that
they
share
the
same
submonoids
of
roots
of
unity.
Indeed,
étale-like
versions
of
this
“essential
Frobenius-like
data”
of
(a
t−F
)
and
(b
t−F
)
are
discussed
in
[IUTchII],
Corollaries
1.10,
1.12;
[IUTchIII],
Theorem
2.2,
(ii)
[cf.
the
data
“(a
v
),
(b
v
),
(c
v
),
(d
v
)”
of
loc.
cit.].
Thus,
from
the
point
of
view
of
studying
inter-universal
Teichmüller
theory,
one
may
essentially
omit
the
detailed
study
of
[EtTh],
§3,
§4,
§5,
either
by
accepting
the
construction
of
the
data
(a
t−F
)
and
(b
t−F
)
“on
faith”
or
by
re-
garding
this
data
as
data
constructed
from
the
scheme-theoretic
objects
discussed
in
[EtTh],
§1,
§2.
§
3.6.
Galois
evaluation,
labels,
symmetries,
and
log-shells
In
the
present
§3.6,
we
discuss
the
theory
of
Galois
evaluation
of
the
κ-coric
rational
functions
and
theta
functions
of
§3.4,
(ii),
(iii).
Here,
we
remark
that
the
term
“Galois
evaluation”
refers
to
the
passage
abstract
functions
→
values
by
first
passing
from
Frobenius-like
—
that
is
to
say,
in
essence,
[pseudo-]monoid-
theoretic
—
versions
of
these
functions
[cf.
the
discussion
of
§3.4,
(ii),
(iii);
§3.5]
to
étale-like
versions
of
these
functions
via
various
forms
of
Kummer
theory
as
dis-
cussed
in
§3.4,
(ii),
(iii),
then
evaluating
these
étale-like
functions
by
restricting
them
to
decomposition
subgroups
[that,
say,
arise
from
closed
points
of
the
curve
under
consideration]
of
the
[possibly
tempered]
arithmetic
fundamental
group
under
consider-
ation
to
obtain
étale-like
versions
of
the
values
of
interest,
and
finally
applying
the
88
Shinichi
Mochizuki
Kummer
theory
of
the
constant
base
field
[i.e.,
as
discussed
in
Example
2.12.1]
to
obtain
Frobenius-like
versions
of
the
values
of
interest
[cf.
Fig.
3.11
below;
[IUTchII],
Remark
1.12.4].
In
fact,
it
is
essentially
a
tautology
that
the
only
way
to
construct
an
assignment
“abstract
functions
→
values”
that
is
compatible
with
the
operation
of
forming
Kummer
classes
is
precisely
by
applying
[some
variant
of]
this
technique
of
Galois
evaluation
[cf.
the
discussion
of
[IUTchII],
Remark
1.12.4].
Moreover,
it
is
inter-
esting
in
this
context
to
observe
[cf.
the
discussion
of
[IUTchII],
Remark
1.12.4]
that
the
well-known
Section
Conjecture
of
anabelian
geometry
—
which,
at
least
historically,
was
expected
to
be
related
to
diophantine
geometry
[cf.
the
discussion
of
[IUTchI],
§I5]
—
suggests
strongly
that,
when
one
applies
the
technique
of
Galois
evaluation,
in
fact,
the
only
suitable
subgroups
of
the
[possibly
tempered]
arithmetic
fundamen-
tal
group
under
consideration
for
the
operation
of
“evaluation”
are
precisely
the
decomposition
subgroups
that
arise
from
the
closed
points
of
the
curve
under
consideration!
From
this
point
of
view,
it
is
also
of
interest
to
observe
that,
in
the
context
of
the
evaluation
of
theta
functions
at
torsion
points
[cf.
(ii)
below],
it
will
be
necessary
to
apply
a
certain
“combinatorial
version
of
the
Section
Conjecture”
[cf.
[IUTchI],
Remark
2.5.1;
the
proof
of
[IUTchII],
Corollary
2.4,
(i)].
Finally,
we
remark
that,
in
order
to
give
a
precise
description
of
the
Galois
evaluation
operations
that
are
performed
in
inter-universal
Teichmüller
theory,
it
will
be
necessary
to
consider,
in
substantial
detail,
·
the
labels
of
the
points
at
which
the
functions
are
to
be
evaluated
[i.e.,
the
points
that
give
rise
to
the
decomposition
subgroups
mentioned
above],
·
the
symmetries
that
act
on
these
labels
[cf.
§3.3,
(v)],
and
·
the
log-shells
that
serve
as
containers
for
the
values
that
are
constructed.
(i)
Passage
to
the
étale-picture,
combinatorial
uni-/multiradiality
of
symmetries:
Recall
from
the
discussion
of
§3.3,
(vi),
that
the
D-Θ
±ell
N
F
-Hodge
theaters
associated
to
the
Θ
±ell
N
F
-Hodge
theaters
“•”
in
the
log-theta-lattice
are
ver-
tically
coric.
That
is
to
say,
one
may
think
of
a
D-Θ
±ell
N
F
-Hodge
theater,
considered
up
to
an
indeterminate
isomorphism,
as
an
invariant
of
each
vertical
line
of
the
log-theta-lattice.
Moreover,
the
étale-like
portion
[i.e.,
the
“G
v
’s”]
of
the
data
of
(a
Θ
),
or,
equivalently,
(a
q
),
of
§3.3,
(vii),
again
considered
up
to
an
indeterminate
isomor-
phism,
may
be
thought
of
as
an
object
constructed
from
the
D-Θ
±ell
N
F
-Hodge
theater
associated
to
the
Θ
±ell
N
F
-Hodge
theater
“•”
under
consideration.
In
particular,
if
one
takes
the
radial
data
to
consist
of
the
D-Θ
±ell
N
F
-Hodge
theater
as-
sociated
to
some
vertical
line
of
the
log-theta-lattice
[considered
up
to
an
in-
determinate
isomorphism!],
the
coric
data
to
consist
of
the
étale-like
portion
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
Frobenius-like
version
of
functions
.
evalu-
..
ation
Frobenius-like
version
of
values
=⇒
étale-like
version
of
functions
evalu-
⇓
ation
Kummer
−1
étale-like
version
of
values
Kummer
⇐=
89
Fig.
3.11:
The
technique
of
Galois
evaluation
[again
considered
up
to
an
indeterminate
isomorphism!]
of
the
data
of
(a
Θ
),
or,
equivalently,
(a
q
),
of
§3.3,
(vii),
and
the
radial
algorithm
to
be
the
assignment
[i.e.,
“construction”]
of
the
above
discussion,
then
one
obtains
a
radial
environ-
ment,
shown
in
Fig.
3.12
below,
that
is
[“tautologically”!]
multiradial
[cf.
[IUTchII],
Corollary
4.11;
[IUTchII],
Fig.
4.3].
[Indeed,
this
multiradial
environment
may
be
thought
of
as
being
simply
a
slightly
more
complicated
version
of
the
multiradial
environment
of
Example
3.2.2,
(ii).]
The
diagram
obtained
by
including,
in
the
diagram
of
Fig.
3.12,
not
just
two
collections
of
radial
data
[that
arise,
say,
from
two
adjacent
vertical
lines
of
the
log-theta-lattice],
but
rather
the
collections
of
radial
data
that
arise
from
all
of
the
vertical
lines
of
the
log-theta-lattice
is
referred
to
as
the
étale-picture
[cf.
[IUTchII],
Fig.
4.3].
Despite
its
tautological
nature,
the
multiradiality
—
i.e.,
permutability
of
D-Θ
±ell
N
F
-Hodge
theaters
as-
sociated
to
distinct
vertical
lines
of
the
log-theta-lattice
—
of
the
étale-
picture
is
nonetheless
somewhat
remarkable
since
[prior
to
passage
to
the
étale-picture!]
the
log-theta-lattice
does
not
admit
symmetries
that
permute
distinct
vertical
lines
of
the
log-theta-lattice.
Next,
we
consider
the
respective
F
±
l
-
and
F
l
-symmetries
of
the
constituent
D-
Θ
±ell
N
F
-Hodge
theaters
[cf.
§3.3,
(v)].
In
this
context,
it
is
useful
to
introduce
sym-
bols
“ ”
and
“>”:
·
“ ”
denotes
the
entire
set
F
l
that
appears
in
the
discussion
of
Fig.
3.8
in
§3.3,
(v),
i.e.,
the
notation
“[.
.
.]”
in
the
upper
left-hand
corner
of
Fig.
3.8
[cf.
[IUTchI],
Fig.
6.5].
·
“>”
denotes
the
entire
set
F
l
that
appears
in
the
discussion
of
Fig.
3.8
in
Shinichi
Mochizuki
90
D-Θ
±ell
N
F
-Hodge
theater
in
the
domain
of
the
Θ-link
radial
algorithm
D-Θ
±ell
N
F
-Hodge
theater
in
the
codomain
of
the
Θ-link
radial
algorithm
étale-like
portion
“G
v
”
of
(a
Θ
),
(a
q
)
Fig.
3.12:
The
multiradiality
of
D-Θ
±ell
N
F
-Hodge
theaters
§3.3,
(v),
i.e.,
the
notation
“[.
.
.]”
in
the
upper
right-hand
corner
of
Fig.
3.8
[cf.
[IUTchI],
Fig.
6.5].
·
The
gluing
shown
in
Fig.
3.8
may
be
thought
of
as
an
assignment
that
sends
0,
→
>
[cf.
[IUTchI],
Proposition
6.7;
[IUTchI],
Fig.
6.5;
the
discussion
of
[IUTchII],
Remark
3.8.2,
(ii);
the
symbol
“ ”
of
[IUTchII],
Corollary
4.10,
(i)].
·
Ultimately,
we
shall
be
interested
in
computing
weighted
averages
of
log-
def
volumes
at
the
various
labels
in
F
l
or
F
l
(⊆
|F
l
|
=
F
l
∪
{0})
[cf.
[IUTchI],
Remark
5.4.2;
the
computations
of
[IUTchIV],
§1].
From
this
point
of
view,
it
is
natural
to
think
in
terms
of
formal
sums
with
rational
coefficients
[0],
def
def
1
2
([j]
+
[−j]),
1
l
([0]
+
[1]
+
[−1]
+
.
.
.
+
[l
]
+
[−l
]),
1
1
l
([
|1|
]
+
.
.
.
+
[
|l
|
])
=
2l
([1]
+
[−1]
+
.
.
.
+
[l
]
+
[−l
])
[ ]
=
[>]
=
def
[
|j|
]
=
—
where
the
“j”
and
“.
.
.”
indicate
arguments
that
range
within
the
positive
integers
between
1
and
l
=
12
(l
−
1).
Note
that
these
assignments
of
formal
sums
are
compatible
with
the
gluing
“0,
→
>”,
i.e.,
relative
to
which
[0]
→
[>],
[ ]
→
[>],
1
(l
+1)
([0]
+
[
|1|
]
+
.
.
.
+
[
|l
|
])
→
[>]
—
where
we
note
that
such
relations
may
be
easily
verified
by
observing
that
the
coefficients
of
“[j]”
and
“[−j]”
always
coincide
and
are
independent
of
j;
thus,
these
relations
may
be
verified
by
substituting
a
single
indeterminate
“w”
for
all
of
the
symbols
“[0]”,
“[j]”,
and
“[−j]”.
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
91
0
Fig.
3.13:
Combinatorial
multiradiality
of
F
l
-symmetries
±
±
±
±
±
±
±
±
!
0
Fig.
3.14:
Combinatorial
uniradiality
of
F
±
l
-symmetries
If
one
extracts
from
the
étale-picture
the
various
F
l
-symmetries
of
the
respective
D-
±ell
Θ
N
F
-Hodge
theaters,
then
one
obtains
a
diagram
as
in
Fig.
3.13
above,
i.e.,
a
diagram
of
various
distinct,
independent
F
l
-actions
that
are
“glued
together
at
a
common
symbol
0”
[cf.
the
discussion
of
[IUTchII],
Remark
4.7.4;
[IUTchII],
Fig.
4.2].
This
diagram
may
be
thought
of
as
a
sort
of
combinatorial
prototype
for
the
phe-
nomenon
of
multiradiality.
On
the
other
hand,
if
one
extracts
from
the
étale-picture
±ell
N
F
-Hodge
theaters,
then
one
the
various
F
±
l
-symmetries
of
the
respective
D-Θ
obtains
a
diagram
as
in
Fig.
3.14
above,
i.e.,
a
diagram
of
various
mutually
interfer-
ing
F
±
l
-actions
that
interfere
with
one
another
as
a
consequence
of
the
fact
that
they
are
“glued
together
at
a
common
symbol
0”
[cf.
the
discussion
of
[IUTchII],
Remark
4.7.4;
[IUTchII],
Fig.
4.1].
This
diagram
may
be
thought
of
as
a
sort
of
combinatorial
prototype
for
the
phenomenon
of
uniradiality.
Finally,
in
this
context,
it
is
also
of
interest
to
observe
that,
if,
in
accordance
with
the
point
of
view
of
the
discussion
of
§2.14,
one
thinks
of
F
l
as
a
sort
of
finite
discrete
approximation
of
“Z”
[cf.
[IUTchI],
Remark
6.12.3,
(i);
[IUTchII],
Remark
4.7.3,
(i)],
and
one
thinks
“Z”
as
the
value
group
of
the
various
completions
[say,
for
simplicity,
at
v
∈
V
non
]
of
K,
then
the
F
l
-symmetry
corresponds
to
a
symmetry
that
only
involves
the
non-unit
portions
of
these
value
groups
at
various
v
∈
V
non
,
while
the
F
±
l
-symmetry
92
Shinichi
Mochizuki
is
a
symmetry
that
involves
a
sort
of
“juggling”
between
local
unit
groups
and
local
value
groups.
This
point
of
view
is
consistent
with
the
fact
[cf.
(iii)
below;
Example
2.12.3,
(v);
§3.3,
(ii),
(vii);
§3.4,
(ii)]
that
the
F
l
-symmetry
is
related
only
to
the
Kummer
theory
surrounding
the
global
value
group
data
(c
Θ
),
while
[cf.
(ii)
below;
Example
2.12.3,
(v);
§3.3,
(ii),
(vii);
§3.4,
(iii),
(iv)]
the
F
±
l
-symmetry
is
related
to
both
the
[local]
Θ
unit
group
data
(a
)
and
the
local
value
group
data
(b
Θ
),
which
are
“juggled”
about
by
the
log-links
of
the
log-theta-lattice.
bad
.
(ii)
Theta
values
and
local
diagonals
via
the
F
±
l
-symmetry:
Let
v
∈
V
Write
⊆
Π
cor
Π
v
⊆
Π
±
v
v
[cf.
[IUTchII],
Definition
2.3,
(i)]
for
the
inclusions
of
tempered
arithmetic
fundamental
groups
[for
suitable
choices
of
basepoints]
determined
by
the
finite
étale
coverings
X
v
→
X
v
→
C
v
[cf.
§3.3,
(i),
(vi);
the
notational
conventions
discussed
at
the
beginning
of
§3.4,
(iii)].
These
tempered
arithmetic
fundamental
groups
of
hyperbolic
orbicurves
⊆
Δ
cor
for
the
over
K
v
admit
natural
outer
surjections
to
G
v
;
write
Δ
v
⊆
Δ
±
v
v
±
cor
respective
kernels
of
these
surjections.
In
fact,
Π
v
and
Π
v
,
together
with
the
above
inclusions,
may
be
reconstructed
functorially
from
the
topological
group
Π
v
[cf.
[EtTh],
±ell
N
F
-Hodge
theater
[cf.
§3.3,
(v)]
Proposition
2.4].
The
F
±
l
-symmetries
of
a
Θ
±
induce
outer
automorphisms
of
Π
v
.
Indeed,
these
outer
automorphisms
may
be
thought
cor
by
the
quotient
of
as
the
outer
automorphisms
of
Π
±
v
induced
by
conjugation
in
Π
v
∼
±
cor
±
±
→
Π
cor
group
Π
v
/Π
v
,
which
admits
a
natural
outer
isomorphism
F
l
v
/Π
v
[cf.
[IUTchII],
Corollary
2.4,
(iii)].
Moreover,
since
these
outer
automorphisms
of
Π
±
v
arise
from
K-linear
automorphisms
of
the
hyperbolic
curve
X
K
[cf.
the
discussion
of
§3.3,
(v);
[IUTchII],
Corollary
2.4,
(iii)],
let
us
observe
that
the
outer
automorphisms
of
Π
±
v
under
consideration
may,
in
fact,
be
thought
±
of
as
Δ
±
v
-outer
automorphisms
of
Π
v
[i.e.,
automorphisms
defined
up
to
composition
with
an
inner
automorphism
induced
by
conjugation
by
an
element
cor
of
Δ
±
v
]
induced
by
conjugation
by
elements
of
Δ
v
.
Next,
let
us
recall
from
the
discussion
of
§3.3,
(v),
concerning
the
F
±
l
-symmetry
that
±
elements
of
F
l
may
be
thought
of
—
up
to
F
l
-indeterminacies
that
may
in
fact,
as
a
consequence
of
the
structure
of
a
Θ
±ell
N
F
-Hodge
theater,
be
synchronized
in
a
fashion
that
is
independent
of
the
choice
of
v
∈
V
bad
[cf.
[IUTchI],
Remark
6.12.4,
(i),
(ii),
(iii)]
—
as
labels
of
cusps
of
X
v
.
Moreover,
such
cusps
of
X
v
may
be
thought
of,
by
applying
a
suitable
functorial
group-theoretic
algorithm,
as
certain
conjugacy
classes
of
subgroups
[i.e.,
cuspidal
inertia
subgroups]
of
Π
±
v
[cf.
[IUTchI],
Definition
6.1,
(iii)].
In
particular,
the
above
observation
implies
that,
if
we
think
of
G
v
as
a
quotient
of
one
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
93
cor
of
the
tempered
arithmetic
fundamental
groups
Π
v
,
Π
±
discussed
above,
and
we
v
,
Π
v
consider
copies
of
this
quotient
G
v
equipped
with
labels
(G
v
)
t
—
where
we
think
of
t
∈
F
l
as
a
conjugacy
class
of
cuspidal
inertia
subgroups
of
Π
±
v
—
then
cor
the
action
of
the
F
±
l
-symmetry
[i.e.,
by
conjugation
in
Π
v
]
on
these
labeled
quotients
{(G
v
)
t
}
t∈F
l
induces
symmetrizing
isomorphisms
between
these
labeled
quotients
that
are
free
of
any
inner
automorphism
indetermina-
cies
[cf.
[IUTchII],
Corollary
3.5,
(i);
[IUTchII],
Remark
3.5.2,
(iii);
[IUTchII],
Remark
4.5.3,
(i)].
The
existence
of
these
symmetrizing
isomorphisms
is
a
phenomenon
that
is
sometimes
referred
to
as
conjugate
synchronization.
Note
that
this
sort
of
situation
differs
radically
from
the
situation
that
arises
for
the
isomorphisms
induced
by
conjugation
def
in
G
K
=
Gal(F
/K)
between
the
various
decomposition
groups
of
v
[that
is
to
say,
copies
of
“G
v
”],
i.e.,
isomorphisms
which
are
only
well-defined
up
to
composition
with
some
indeterminate
inner
automorphism
of
the
decomposition
group
under
con-
sideration
[cf.
the
discussion
of
[IUTchII],
Remark
2.5.2,
(iii)].
Relative
to
the
theme
of
“synchronization”,
another
important
role
played
by
the
F
±
l
-symmetry
is
the
role
of
synchronizing
the
±-indeterminacies
that
occur
at
the
portions
labeled
by
various
valuations
∈
V
on
the
“left-hand
side”
[cf.
Fig.
3.8]
of
a
Θ
±ell
N
F
-Hodge
theater
[cf.
[IUTchII],
Remark
4.5.3,
(iii)].
In
this
context,
since
tempered
arithmetic
fundamental
groups,
unlike
conventional
profinite
étale
fundamental
groups,
are
only
defined
at
a
specific
v
∈
V
bad
,
one
technical
issue
that
arises,
when
one
considers
the
task
of
relating
the
symmetrizing
isomorphisms
discussed
above
at
different
v
∈
V
bad
[or,
indeed,
to
the
theory
at
valuations
∈
V
good
]
is
the
issue
of
comparing
tempered
and
profinite
conjugacy
classes
of
various
types
of
subgroups
[i.e.,
such
as
cuspidal
inertia
groups]
—
an
issue
that
is
resolved
[cf.
the
application
of
[IUTchI],
Corollary
2.5,
in
the
proof
of
[IUTchII],
Corollary
2.4]
by
applying
the
theory
of
[Semi].
The
symmetrizing
isomorphisms
discussed
above
may
be
applied
not
only
to
copies
of
the
étale-like
object
G
v
but
also
to
various
Frobenius-like
objects
that
are
“closely
re-
lated”
to
G
v
[cf.
the
pairs
“G
k
O
k
”
of
Example
2.12.1;
[IUTchII],
Corollary
3.6,
(i)].
Moreover,
an
analogous
theory
of
symmetrizing
isomorphisms
may
be
developed
at
val-
uations
∈
V
good
[cf.
[IUTchII],
Corollary
4.5,
(iii);
[IUTchII],
Corollary
4.6,
(iii)].
The
graphs,
or
diagonals,
of
these
symmetrizing
isomorphisms
at
various
valuations
∈
V
Shinichi
Mochizuki
94
may
be
thought
of
as
corresponding
to
the
symbol
“ ”
discussed
in
(i),
or
indeed,
after
applying
the
gluing
that
appears
in
the
structure
of
a
Θ
±ell
N
F
-Hodge
theater
[cf.
(i);
§3.3,
(v)],
to
the
symbols
“0”,
“>”
[cf.
[IUTchII],
Corollary
3.5,
(iii);
[IUTchII],
Corol-
lary
3.6,
(iii);
[IUTchII],
Corollary
4.5,
(iii);
[IUTchII],
Corollary
4.6,
(iii);
[IUTchII],
Corollary
4.10,
(i)].
Moreover,
the
data
labeled
by
these
symbols
“ ”,
“0”,
“>”,
form
the
data
that
is
ulti-
mately
actually
used
in
the
horizontally
coric
unit
group
portion
(a
Θ
),
(a
q
)
[cf.
§3.3,
(vii)]
of
the
data
in
the
codomain
and
domain
of
the
Θ-link
[cf.
[IUTchIII],
Theorem
1.5,
(iii)].
The
significance
of
this
approach
to
constructing
the
data
of
(a
Θ
),
(a
q
)
lies
in
the
fact
that
the
“descent”
[cf.
[IUTchIII],
Remark
1.5.1,
(i)]
from
the
individual
labels
t
∈
F
l
to
the
symbols
“ ”/“0”/“>”
that
is
effected
by
the
various
symmetrizing
isomorphisms
gives
rise
to
horizontally
coric
data
—
i.e.,
data
that
is
shared
by
the
codomain
and
domain
of
the
Θ-link
—
that
serves
as
a
container
[cf.
the
discussion
of
(iv)
below]
for
the
various
theta
values
[well-defined
up
to
multiplication
by
a
2l-th
root
of
unity]
q
j
2
—
where
we
think
of
j
=
1,
.
.
.
,
l
as
corresponding
to
an
element
of
F
l
obtained
×
⊆
F
l
—
obtained
by
Galois
evaluation
by
identifying
two
elements
±t
∈
F
l
[cf.
[IUTchII],
Corollary
2.5;
[IUTchII],
Remark
2.5.1;
[IUTchII],
Corollary
3.5,
(ii);
[IUTchII],
Corollary
3.6,
(ii)],
i.e.,
by
restricting
the
Kummer
classes
of
the
[recipro-
cals
of
l-th
roots
of]
theta
functions
on
Ÿ
v
discussed
in
§3.4,
(iii)
[cf.
also
the
data
“(a
t−F
)”
discussed
in
§3.5]
¨
in
the
open
·
first
to
the
decomposition
groups,
denoted
by
the
notation
“
”,
subgroup
of
Π
v
corresponding
to
Ÿ
v
determined
[up
to
conjugation
in
Π
v
—
cf.
[IUTchII],
Proposition
2.2;
[IUTchII],
Corollary
2.4]
by
the
connected
—
i.e.,
so
as
not
to
give
rise
to
distinct
basepoints
for
distinct
labels
j
=
1,
.
.
.
,
l
[cf.
the
discussion
of
[IUTchII],
Remarks
2.6.1,
2.6.2,
2.6.3]
—
“line
segment”
of
labels
of
irreducible
components
of
the
special
fiber
of
Ÿ
v
{−l
,
−l
+
1,
.
.
.
,
−1,
0,
1,
.
.
.
,
l
−
1,
l
}
⊆
Z
[cf.
Fig.
3.9
and
the
surrounding
discussion;
[IUTchII],
Remark
2.1.1,
(ii)];
and
·
then
to
the
decomposition
groups
associated
to
“evaluation
points”
—
i.e.,
cusps
translated
by
a
zero-labeled
evaluation
point
—
labeled
by
±t
∈
F
×
l
⊆
F
l
.
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
95
Finally,
we
remark
that
the
Kummer
theory
that
relates
the
corresponding
étale-like
and
Frobenius-like
data
that
appears
in
the
various
symmetrizing
isomorphisms
just
discussed
only
involves
local
data,
i.e.,
the
data
of
(a
Θ
)
and
(b
Θ
),
hence
[cf.
the
discus-
sion
of
§3.4,
(v);
Fig.
3.10]
is
compatible
with
the
topologies
of
the
various
tempered
or
profinite
Galois
or
arithmetic
fundamental
groups
involved.
The
significance
of
this
compatibility
with
topologies
lies
in
the
fact
it
means
that
the
Kummer
isomorphisms
that
appear
may
be
computed
relative
to
some
finite
étale
covering
of
the
schemes
involved,
i.e.,
relative
to
a
situation
in
which
—
unlike
the
situation
that
arises
if
one
considers
some
sort
of
projective
limit
of
multiplicative
monoids
associated
to
rings
—
the
ring
structure
of
the
schemes
involved
is
still
intact.
That
is
to
say,
since
the
log-link
is
defined
by
applying
the
formal
power
series
of
the
natural
logarithm,
an
object
that
can
only
be
defined
if
both
the
additive
and
the
multiplicative
structures
of
the
[topological]
rings
involved
are
available,
this
computability
allows
one
to
compare
—
hence
to
establish
the
com-
patibility
of
—
the
various
symmetrizing
isomorphisms
just
discussed
in
the
codomain
and
domain
of
the
log-link
[cf.
[IUTchII],
Remark
3.6.4,
(i);
[IUTchIII],
Remark
1.3.2;
the
discussion
of
Step
(vi)
of
the
proof
of
[IUTchIII],
Corollary
3.12].
This
compatibility
plays
an
important
role
in
inter-universal
Teichmüller
theory.
(iii)
Number
field
values
and
global
diagonals
via
the
F
l
-symmetry:
We
begin
by
considering
certain
field
extensions
of
the
field
F
mod
:
write
·
F
sol
⊆
F
for
the
maximal
solvable
extension
of
F
mod
in
F
[cf.
[IUTchI],
Definition
3.1,
(b)];
·
F
(μ
l
,
C
F
)
for
the
field
obtained
by
adjoining
to
the
function
field
of
C
F
the
l-th
roots
of
unity
[cf.
the
field
“F
(μ
l
)
·
L
C
”
of
[IUTchI],
Remark
3.1.7,
(iii)];
·
F
(μ
l
,
κ-sol)
for
the
field
obtained
by
adjoining
to
F
(μ
l
,
C
F
)
arbitrary
roots
of
F
sol
-multiples
of
κ-coric
rational
functions
[cf.
§3.4,
(ii)]
in
F
(μ
l
,
C
F
)
[cf.
the
field
“F
(μ
l
)
·
L
C
(κ-sol)”
of
[IUTchI],
Remark
3.1.7,
(iii)];
·
F
(C
K
)
for
the
Galois
closure
over
the
field
F
(μ
l
,
C
F
)
of
the
function
field
of
C
K
[cf.
the
field
“L
C
(C
K
)”
of
[IUTchI],
Remark
3.1.7,
(iii)].
Then
one
verifies
immediately,
by
applying
the
fact
that
the
finite
group
SL
2
(F
l
)
[where
we
recall
from
§3.3,
(i),
that
l
≥
5]
is
perfect,
that
F
(μ
l
,
κ-sol)
and
F
(C
K
)
are
linearly
disjoint
over
F
(μ
l
,
C
F
)
[cf.
[IUTchI],
Remark
3.1.7,
(iii)].
It
then
follows,
in
an
essentially
formal
way,
from
this
linear
disjointness
[cf.
[IUTchI],
Shinichi
Mochizuki
96
Remark
3.1.7,
(ii),
(iii);
[IUTchI],
Example
5.1,
(i),
(v);
[IUTchI],
Remark
5.1.5;
[IUTchII],
Corollary
4.7,
(i),
(ii);
[IUTchII],
Corollary
4.8,
(i),
(ii)]
that:
·
the
various
elements
in
F
mod
or
F
sol
may
be
obtained
by
Galois
evaluation,
i.e.,
by
restricting
the
Kummer
classes
of
the
κ-coric
rational
functions
discussed
in
§3.4,
(ii),
to
the
various
decomposition
groups
that
arise,
respectively,
from
F
mod
-
or
F
sol
-rational
points;
·
this
construction
of
F
mod
or
F
sol
via
Galois
evaluation
may
be
done
in
a
fashion
that
is
compatible
with
the
labels
∈
F
l
and
the
F
l
-symmetry
that
appear
in
a
Θ
±ell
N
F
-Hodge
theater
[cf.
the
discussion
of
§3.3,
(v);
the
right-hand
side
of
Fig.
3.8];
·
in
particular,
this
compatibility
with
labels
and
the
F
l
-symmetry
induces
symmetrizing
isomorphisms
between
copies
of
F
mod
or
F
sol
that
determine
graphs,
or
diagonals,
which
may
be
thought
of
as
corresponding
to
the
sym-
bol
“>”
[cf.
the
discussion
of
(i),
(ii)].
In
this
context,
we
note
that
this
approach
to
constructing
elements
of
NF’s
by
re-
stricting
Kummer
classes
of
rational
functions
on
hyperbolic
curves
to
decomposition
groups
of
points
defined
over
an
NF
is
precisely
the
approach
taken
in
the
functorial
algorithms
of
[AbsTopIII],
Theorem
1.9
[cf.,
especially,
[AbsTopIII],
Theorem
1.9,
(d)].
Also,
we
observe
that,
although
much
of
the
above
discussion
runs
in
a
somewhat
paral-
lel
fashion
to
the
discussion
in
(ii)
of
the
F
±
l
-symmetry
and
the
construction
of
theta
values
via
Galois
evaluation,
there
are
important
differences,
as
well,
between
the
theta
and
NF
cases
[cf.
[IUTchIII],
Remark
2.3.3]:
·
First
of
all,
the
symmetrizing
isomorphisms/diagonals
associated
to
the
F
l
-symmetry
are
not
compatible
with
the
symmetrizing
isomorphisms/diagonals
associated
to
the
F
±
l
-symmetry,
except
on
the
respective
restrictions
of
these
two
collections
of
symmetrizing
isomorphisms
to
copies
of
F
mod
[cf.
[IUTchII],
Remark
4.7.2].
Moreover,
for
various
technical
reasons
related
to
conjugate
synchronization,
it
is
of
fundamental
importance
in
the
theory
to
isolate
the
±
F
l
-symmetry
from
the
F
l
-symmetry
[cf.
the
discussion
of
[IUTchII],
Remarks
2.6.2,
4.7.3,
4.7.5,
4.7.6].
·
Unlike
the
Kummer
theory
applied
in
the
theta
case,
the
Kummer
theory
applied
in
the
NF
case
is
not
compatible
with
the
topologies
of
the
various
profinite
Galois
or
arithmetic
fundamental
groups
that
appear
[cf.
the
discus-
sion
of
§3.4,
(v)].
On
the
other
hand,
this
will
not
cause
any
problems
since
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
97
there
is
no
issue,
in
the
NF
case,
of
applying
formal
power
series
such
as
the
power
series
of
the
natural
logarithm
[cf.
the
final
portion
of
the
discussion
of
(ii);
[IUTchIII],
Remark
2.3.3,
(vii),
(viii)].
·
In
the
Galois
evaluation
applied
in
the
theta
case,
one
is
concerned
with
con-
structing,
at
a
level
where
the
arithmetic
holomorphic
structure
[i.e.,
the
ring
structure]
is
still
intact,
theta
values
that
depend,
in
an
essential
way,
on
the
label
“j”.
By
contrast,
in
the
Galois
evaluation
applied
in
the
NF
case,
one
only
constructs,
at
such
a
level
where
the
arithmetic
holomorphic
structure
×
is
still
intact,
the
totality
of
[various
copies
of]
the
multiplicative
monoid
F
mod
associated
to
the
number
field
F
mod
:
that
is
to
say,
a
dependence
on
the
label
“j”
only
appears
at
the
level
of
the
mono-analytic
structures
constituted
by
the
global
realified
Frobenioids
of
(c
Θ
),
i.e.,
in
the
form
of
a
sort
of
ratio,
or
weight,
“j
2
”
[cf.
the
fourth
display
of
[IUTchII],
Corollary
4.5,
(v)]
between
the
arithmetic
degrees
at
the
label
“j”
and
the
arithmetic
degrees
at
the
label
“1”
[cf.
[IUTchIII],
Remark
2.3.3,
(iii);
[IUTchIII],
Remark
3.11.4,
(i)].
In
the
context
of
this
final
difference
between
the
theta
and
NF
cases,
it
is
perhaps
of
2
×
×
×
)
j
(⊆
F
mod
)
of
F
mod
is
interest
to
observe
that
a
similar
sort
of
“weighted
copy”
(F
mod
not
possible
at
the
level
of
arithmetic
holomorphic
structures
[that
is
to
say,
in
the
sense
that
it
is
not
compatible
with
the
additive
interpretation
of
line
bundles,
i.e.,
in
2
×
×
)
j
(⊆
F
mod
)
is
not
closed
under
terms
of
modules]
since
this
“weighted
copy”
(F
mod
addition
[cf.
the
discussion
of
the
final
portion
of
§3.3,
(iv);
[AbsTopIII],
Remark
5.10.2,
(iv)].
(iv)
Actions
on
log-shells:
At
this
point,
the
reader
may
have
noticed
two
ap-
parent
shortcomings
[which
are,
in
fact,
closely
related!]
in
the
theory
of
Galois
eval-
uation
developed
thus
far
in
the
present
§3.6:
·
Unlike
the
case
with
the
Kummer
theory
of
κ-coric
rational
functions
and
theta
functions/mono-theta
environments
discussed
in
§3.4,
the
discussion
of
Galois
evaluation
given
above
does
not
mention
any
multiradiality
properties.
·
Ultimately,
one
is
interested
in
relating
[the
Kummer
theory
surrounding]
Frobenius-like
structures
—
such
as
the
theta
values
and
copies
of
NF’s
that
arise
from
Galois
evaluation
—
in
the
domain
of
the
Θ-link
to
[the
Kummer
theory
surrounding]
Frobenius-like
structures
in
the
codomain
of
the
Θ-link,
i.e.,
in
accordance
with
the
discussion
of
the
technique
of
mono-anabelian
transport
in
§2.7,
§2.9.
On
the
other
hand,
since
the
theta
values
and
copies
of
NF’s
that
arise
from
Galois
evaluation
are
not
[necessarily]
local
units
at
the
various
v
∈
V,
it
is
by
no
means
clear
how
to
relate
this
Galois
evaluation
output
data
to
the
codomain
of
the
Θ-link
using
the
horizontally
coric
portion
—
i.e.,
the
[local]
unit
group
portion
(a
Θ
)
and
(a
q
)
—
of
the
Θ-link.
98
Shinichi
Mochizuki
In
fact,
these
two
shortcomings
are
closed
related:
That
is
to
say,
the
existence
of
the
obstruction
discussed
in
§3.4,
(i),
to
the
approach
to
Kummer
theory
taken
in
Example
2.12.1,
(ii),
that
arises
from
the
natural
action
“O
k
×μ
Z
×
”
implies
—
in
light
of
the
nontrivial
extension
structure
that
exists
between
the
value
groups
and
units
of
finite
subextensions
of
“k”
in
“k”
[cf.
the
discussion
in
the
final
portion
of
Example
2.12.1,
(iii)]
—
that,
at
least
in
any
sort
of
a
priori
or
natural
sense,
the
output
data
—
i.e.,
theta
values
and
copies
of
NF’s
—
of
the
Galois
evaluation
operations
discussed
in
(ii),
(iii)
above
[i.e.,
which
lies
in
various
copies
of
“k”!]
is
by
no
means
multiradial
[cf.
[IUTchII],
Remark
2.9.1,
(iii);
[IUTchII],
Remark
3.4.1,
(ii);
[IUTchII],
Remark
3.7.1;
[IUTchIII],
Remark
2.2.1,
(iv)].
One
of
the
fundamental
ideas
of
inter-universal
Teichmüller
theory
is
that
one
may
apply
the
theory
of
the
log-link
and
log-shells
to
obtain
a
solution
to
these
closely
related
shortcomings.
More
precisely,
from
the
point
of
view
of
the
log-theta-lattice,
the
log-link
from
the
lattice
point
(n,
m
−
1)
to
the
lattice
point
(n,
m)
[where
n,
m
∈
Z]
allows
one
to
construct
[cf.
the
notation
of
Example
2.12.3,
(iv)]
a
holomorphic
Frobenius-like
log-shell
“I”
at
(n,
m)
from
the
“O
k
×μ
”
at
(n,
m
−
1).
Thus,
the
output
data
—
i.e.,
theta
values
and
copies
of
NF’s
—
of
the
Galois
evaluation
operations
discussed
in
(ii),
(iii)
above
at
(n,
m)
acts
naturally
on
the
“I
⊗
Q”
[i.e.,
the
copy
of
“k”]
at
(n,
m)
that
arises
from
this
log-link
from
(n,
m
−
1)
to
(n,
m)
[cf.
[IUTchIII],
Proposition
3.3,
(i);
[IUTchIII],
Proposition
3.4,
(ii);
[IUTchIII],
Definition
3.8,
(ii)].
On
the
other
hand,
this
gives
rise
to
a
fundamental
dilemma:
Since
the
construction
of
the
log-link
—
i.e.,
at
a
more
concrete
level,
the
formal
power
series
of
the
natural
logarithm
—
can
only
be
defined
if
both
the
additive
and
the
multiplicative
structures
of
the
[topological]
rings
involved
are
available,
the
log-link
and
hence,
in
particular,
the
construction
of
log-shells
just
discussed,
at,
say,
the
lattice
point
(n,
m),
are
meaningless,
at
least
in
any
a
priori
sense,
from
the
point
of
view
of
the
lattice
point
(n
+
1,
m),
i.e.,
the
codomain
of
the
Θ-link
from
(n,
m)
to
(n
+
1,
m)
[cf.
[IUTchIII],
Remark
3.11.3;
Steps
(iii)
and
(iv)
of
the
proof
of
[IUTchIII],
Corollary
3.12].
Another
of
the
fundamental
ideas
of
inter-universal
Teichmüller
theory
is
the
following
[cf.
the
discussion
in
§3.3,
(ii),
of
the
symmetry
of
the
[infinite!]
vertical
lines
of
the
log-theta-lattice
with
respect
to
arbitrary
vertical
translations!]:
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
99
By
considering
structures
that
are
invariant
with
respect
to
vertical
shifts
of
the
log-theta-lattice
—
i.e.,
vertically
coric
structures
such
as
holomorphic/
mono-analytic
étale-like
log-shells
that
serve
as
containers
for
Frobenius-like
objects
such
as
holomorphic
Frobenius-like
log-shells
[cf.
Fig.
3.15
below]
or
notions
of
vertical
invariance
such
as
upper
semi-commutativity
or
log-
volume
compatibility
[cf.
Example
2.12.3,
(iv)]
—
and
then
transporting
these
invariant
structures
to
the
opposite
side
of
the
Θ-link
by
means
of
mono-
analytic
Frobenius-like
log-shells
[i.e.,
which
may
be
constructed
directly
from
the
data
“O
k
×μ
”
that
appears
in
the
horizontally
coric
data
(a
Θ
),
(a
q
)
of
the
Θ-link],
one
may
construct
multiradial
containers
for
the
output
data
—
i.e.,
theta
values
and
copies
of
NF’s
—
of
the
Galois
evaluation
operations
discussed
above.
...
→
•→•→•
→
...
...
↓
...
◦
Fig.
3.15:
A
vertical
line
of
the
log-theta-lattice
[shown
horizontally]:
holomorphic
Frobenius-like
structures
“•”
at
each
lattice
point
related,
via
various
Kummer
isomorphisms
[i.e.,
vertical
or
diagonal
arrows],
to
vertically
coric
holomorphic
étale-like
structures
“◦”
Here,
we
observe
that
each
of
the
four
types
of
log-shells
discussed
in
Example
2.12.3,
(iv),
plays
an
indispensable
role
in
the
theory
[cf.
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob7);
[IUTchIII],
Remark
3.12.2,
(iv),
(v)]:
·
the
holomorphic
Frobenius-like
log-shells
satisfy
the
property
[unlike
their
mono-analytic
counterparts!]
that
the
log-link
—
whose
construction
re-
quires
the
use
of
the
topological
ring
structure
on
these
log-shells!
—
may
be
ap-
plied
to
them,
as
well
as
the
property
[unlike
their
étale-like
counterparts!]
that
they
belong
to
a
fixed
vertical
position
of
a
vertical
line
of
the
log-theta-lattice,
hence
are
meaningful
even
in
the
absence
of
the
log-link
and,
in
particular,
may
be
related
directly
to
the
Θ-link;
·
the
holomorphic
étale-like
log-shells
allow
one
[unlike
their
Frobenius-like
counterparts!]
to
relate
holomorphic
Frobenius-like
log-shells
at
different
verti-
cal
positions
of
a
vertical
line
of
the
log-theta-lattice
to
one
another
in
a
fashion
[unlike
their
mono-analytic
counterparts!]
that
takes
into
account
the
log-link
[whose
construction
requires
the
use
of
the
topological
ring
structure
on
these
log-shells!];
·
the
mono-analytic
Frobenius-like
log-shells
satisfy
the
property
[unlike
their
holomorphic
counterparts!]
that
they
may
be
constructed
directly
from
Shinichi
Mochizuki
100
the
data
“O
k
×μ
”
that
appears
in
the
horizontally
coric
data
(a
Θ
),
(a
q
)
of
the
Θ-link,
as
well
as
the
property
[unlike
their
étale-like
counterparts!]
that
they
belong
to
a
fixed
vertical
position
of
a
vertical
line
of
the
log-theta-lattice,
hence
are
meaningful
even
in
the
absence
of
the
log-link
and,
in
particular,
may
be
related
directly
to
the
Θ-link
[cf.
the
discussion
in
the
final
portion
of
§3.3,
(ii)];
·
the
mono-analytic
étale-like
log-shells
satisfy
the
property
[unlike
their
holomorphic
counterparts!]
that
they
may
be
constructed
directly
from
the
data
“G
k
”
that
appears
in
the
horizontally
coric
data
(a
Θ
),
(a
q
)
of
the
Θ-link,
as
well
as
the
property
[unlike
their
Frobenius-like
counterparts,
when
taken
alone!]
that
they
may
be
used,
in
conjunction
with
their
Frobenius-like
counterparts,
to
relate
mono-analytic
Kummer
theory
[i.e.,
Kummer
isomorphisms
between
mono-analytic
Frobenius-like/étale-like
log-shells]
to
holomorphic
Kummer
the-
ory
[i.e.,
Kummer
isomorphisms
between
holomorphic
Frobenius-like/étale-like
log-shells].
In
particular,
the
significance
of
working
with
mono-analytic
Frobenius-like
log-shells
may
be
understood
as
follows.
If,
instead
of
working
with
mono-analytic
Frobenius-
like
log-shells,
one
simply
passes
from
holomorphic
Frobenius-like
log-shells
at
arbitrary
vertical
positions
[in
a
single
vertical
line
of
the
log-theta-lattice]
to
holomorphic
étale-
like
log-shells
[via
Kummer
isomorphisms]
and
then
to
mono-analytic
étale-like
log-shells
[by
forgetting
various
structures]
—
i.e.,
holomorphic
Frobenius-like
log-shells
(Ind3)
(Ind1)
holomorphic
étale-like
log-shells
mono-analytic
étale-like
log-shells
—
then
the
relationship
between
the
mono-analytic
étale-like
log-shells
[whose
vertical
position
is
indeterminate!]
and
the
various
holomorphic
Frobenius-like
log-shells
[in
the
vertical
line
under
consideration]
is
subject
simultaneously
to
indeterminacies
arising
from
both
the
log-link
[i.e.,
“(Ind3)”
—
cf.
the
discussion
of
§3.7,
(i),
below]
and
the
Θ-link
[i.e.,
“(Ind1)”
—
cf.
the
discussion
of
§3.7,
(i),
below].
Relative
to
the
analogy
discussed
in
the
final
portion
of
§3.3,
(ii),
between
the
log-theta-lattice
and
“C
×
\GL
+
(V
)/C
×
”,
such
simultaneous
indeterminacies
correspond
to
indeterminacies
with
respect
to
the
action
of
the
subgroup
of
GL
+
(V
)
generated
by
C
×
and
“
0
t
0
1
”.
By
contrast,
by
stipulating
that
the
passage
from
holomorphic
étale-like
log-shells
to
mono-
analytic
étale-like
log-shells
be
executed
in
conjunction
with
the
Kummer
isomorphisms
[implicit
in
the
data
that
is
glued
together
in
definition
of
the
Θ-link]
with
corresponding
Frobenius-like
log-shells
at
the
vertical
position
“0”
—
i.e.,
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
holomorphic
Frobenius-like
log-shells
holomorphic
étale-like
log-shells
(Ind3)
↑
holomorphic
Frobenius-like
log-shells
at
0
101
mono-analytic
étale-like
log-shells
↑
(Ind1,2)
mono-analytic
Frobenius-like
log-shells
at
0
[where
the
vertical
arrows
denote
the
respective
Kummer
isomorphisms]
—
one
obtains
a
“decoupling”
of
the
log-link/Θ-link
indeterminacies,
i.e.,
·
a
partially
rigid
relationship
between
the
holomorphic/mono-analytic
étale-like
log-shells
and
the
holomorphic/mono-analytic
Frobenius-like
log-shells
at
the
vertical
position
0
[in
the
vertical
line
under
consideration]
that
is
subject
only
to
indeterminacies
arising
from
the
Θ-link
[i.e.,
“(Ind1),
(Ind2)”
—
cf.
the
discussion
of
§3.7,
(i),
below],
together
with
·
a
partially
rigid
relationship
between
the
holomorphic
Frobenius-
like
log-shells
at
the
vertical
position
0
and
the
various
holomorphic
Frobenius-like
log-shells
at
arbitrary
vertical
positions
[in
the
vertical
line
under
consideration],
i.e.,
via
holomorphic
étale-like
log-shells,
that
is
subject
only
to
indeterminacies
arising
from
the
log-link
[i.e.,
“(Ind3)”
—
cf.
the
discussion
of
§3.7,
(i),
below].
Relative
to
the
analogy
discussed
in
the
final
portion
of
§3.3,
(ii),
between
the
log-theta-
lattice
and
“C
×
\GL
+
(V
)/C
×
”,
this
“decoupling”
of
indeterminacies
corresponds
to
an
indeterminacy
with
respect
to
an
action
of
C
×
from
the
left,
together
with
a
distinct
action
of
“
0
t
0
1
”
from
the
right.
(v)
Processions:
In
order
to
achieve
the
multiradiality
discussed
in
(iv)
for
Galois
evaluation
output
data,
one
further
technique
must
be
introduced,
namely,
the
use
of
processions
[cf.
[IUTchI],
Definition
4.10],
which
serve
as
a
sort
of
mono-analytic
substitute
for
the
various
labels
in
F
l
,
|F
l
|
(=
F
l
∪
{0}),
or
F
l
discussed
in
(i).
That
is
to
say,
since
these
labels
are
closely
related
to
the
various
cuspidal
inertia
subgroups
of
the
geometric
fundamental
groups
“Δ”
of
the
hyperbolic
orbicurves
involved
[cf.
the
Shinichi
Mochizuki
102
discussion
of
(i),
(ii);
§2.13;
§3.3,
(v)],
it
follows
that
these
labels
are
not
horizontally
coric
[i.e.,
not
directly
visible
to
the
opposite
side
of
the
Θ-link
—
cf.
the
discussion
of
[IUTchIV],
Remark
3.6.3,
(ii)]
and
indeed
do
not
even
admit,
at
least
in
any
a
priori
sense,
any
natural
multiradial
formulation.
The
approach
taken
in
inter-universal
Teichmüller
theory
to
dealing
with
this
state
of
affairs
[cf.
[IUTchI],
Proposition
6.9]
is
to
consider
the
diagram
of
inclusions
of
finite
sets
S
±
1
→
S
±
1+1=2
→
...
→
S
±
j+1
→
...
→
S
±
1+l
=l
±
—
where
we
write
S
±
j+1
=
{0,
1,
.
.
.
,
j},
for
j
=
0,
.
.
.
,
l
,
and
we
think
of
each
of
these
finite
sets
as
being
subject
to
arbitrary
permutation
automorphisms.
That
is
to
say,
we
think
of
def
the
set
S
±
j+1
as
a
container
for
the
labels
0,
1,
.
.
.
,
j,
and
of
the
label
“j”
as
“some”
element
of
this
container
set,
i.e.,
for
each
j,
there
is
an
indeterminacy
of
j
+
1
possibilities
for
the
element
of
this
container
set
that
corresponds
to
j.
Here,
we
note
in
passing
that
this
sort
of
indeterminacy
is
substantially
milder
than
the
indeterminacies
that
occur
if
one
considers
each
j
only
as
“some”
element
of
S
l
±
,
in
which
case
every
j
is
subject
to
an
indeterminacy
of
l
±
possibilities
—
cf.
[IUTchI],
Proposition
6.9,
(i),
(ii).
One
then
regards
each
such
container
set
as
an
index
set
for
a
collection
—
which
is
referred
to
as
a
“capsule”
[cf.
[IUTchI],
§0]
—
of
copies
of
some
sort
of
étale-like
mono-analytic
prime-strip.
An
étale-like
mono-analytic
prime-strip
is,
roughly
speaking,
a
collection
of
copies
of
data
“G
k
”
indexed
by
v
∈
V
[cf.
[IUTchI],
Fig.
I1.2,
and
the
surrounding
discus-
sion;
[IUTchI],
Definition
4.1,
(iii)].
Now
each
étale-like
mono-analytic
prime-strip
in
a
capsule,
as
described
above,
gives
rise,
at
each
v
∈
V,
to
mono-analytic
étale-like
log-shells,
on
which
the
Galois
evaluation
output
data
acts,
in
the
fashion
described
in
(iv),
up
to
various
indeterminacies
that
arise
from
the
passage
from
holomorphic
Frobenius-like
log-shells
to
mono-
analytic
étale-like
log-shells
[cf.
Figs.
3.16,
3.17
below,
where
each
“/
±
”
denotes
an
étale-like
mono-analytic
prime-strip].
These
indeterminacies
will
be
discussed
in
more
detail
in
§3.7,
(i),
below.
In
fact,
ultimately,
from
the
point
of
various
log-volume
computations,
it
is
more
natural
to
consider
the
Galois
evaluation
output
data
as
acting,
up
to
various
indeterminacies,
on
certain
tensor
products
of
the
various
log-shells
indexed
by
a
particular
container
set
S
±
j+1
.
Such
tensor
products
are
referred
to
as
tensor
packets
[cf.
[IUTchIII],
Propositions
3.1,
3.2].
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
q
1
2
q
j
103
2
q
(l
)
/
±
→
/
±
/
±
→
.
.
.
→
/
±
/
±
.
.
.
/
±
→
.
.
.
→
/
±
/
±
.
.
.
.
.
.
/
±
S
±
S
±
S
±
S
±
1
1+1=2
j+1
1+l
=l
±
Fig.
3.16:
Theta
values
acting
on
tensor
packets
×
×
×
)
1
(F
mod
)
j
(F
mod
)
l
(F
mod
±
±
±
±
±
±
±
±
→
.
.
.
→
/
/
.
.
.
/
→
.
.
.
→
/
/
.
.
.
.
.
.
/
±
/
→
/
/
S
±
S
±
S
±
S
±
1
1+1=2
j+1
1+l
=l
±
×
Fig.
3.17:
Copies
of
F
mod
acting
on
tensor
packets
§
3.7.
Log-volume
estimates
via
the
multiradial
representation
In
the
following,
we
outline
the
statements
of
and
relationships
between
the
main
results
[cf.
[IUTchIII],
Theorem
3.11;
[IUTchIII],
Corollary
3.12;
[IUTchIV],
Theorem
1.10;
[IUTchIV],
Corollaries
2.2,
2.3]
of
inter-universal
Teichmüller
theory.
(i)
Multiradial
representation
of
the
Θ-pilot
object
up
to
mild
inde-
terminacies:
The
content
of
the
discussion
of
§3.4,
§3.5,
and
§3.6
may
be
summarized
as
follows
[cf.
[IUTchIII],
Theorem
A;
[IUTchIII],
Theorem
3.11]:
the
data
in
the
domain
(a
Θ
),
(b
Θ
),
and
(c
Θ
)
[cf.
§3.3,
(vii)]
of
the
Θ-link
may
be
expressed
in
a
fashion
that
is
multiradial,
when
considered
up
to
certain
indeterminacies
(Ind1),
(Ind2),
(Ind3)
[cf.
the
discussion
below],
with
respect
to
the
radial
algorithm
(a
Θ
),
(b
Θ
),
(c
Θ
)
→
(a
Θ
)
[cf.
the
discussion
at
the
beginning
of
§3.4]
by
regarding
this
[Frobenius-like!]
data
(a
Θ
),
(b
Θ
),
and
(c
Θ
)
[up
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)]
as
data
[cf.
Fig.
3.18
below]
that
is
constructed
by
·
first
applying
the
Kummer
theory
and
multiradial
decou-
plings/cyclotomic
rigidity
of
κ-coric
rational
functions
in
the
case
of
(c
Θ
)
[cf.
§3.4,
(ii),
(v)]
and
of
theta
functions/mono-theta
environments
in
the
case
of
(b
Θ
)
[cf.
§3.4,
(iii),
(iv),
(v);
§3.5];
and
·
then
applying
the
theory
of
Galois
evaluation,
log-shells,
and
processions,
together
with
symmetrizing
isomorphisms
at
l-
Θ
torsion
points
via
the
F
±
l
-symmetry,
in
the
case
of
(b
)
[cf.
§3.6,
(i),
(ii),
(iv),
(v)],
and
at
decomposition
groups
corresponding
to
F
mod
-/F
sol
-rational
points
via
the
F
l
-symmetry,
in
the
case
of
Θ
(c
)
[cf.
§3.6,
(i),
(iii),
(iv),
(v)].
Shinichi
Mochizuki
104
q
1
q
j
2
2
/
±
→
/
±
/
±
→
...
→
/
±
/
±
.
.
.
/
±
→
...
→
/
±
/
±
.
.
.
.
.
.
/
±
×
(F
mod
)
1
×
(F
mod
)
j
×
(F
mod
)
l
q
(l
)
Fig.
3.18:
The
full
multiradial
representation
Here,
we
recall
from
§3.6,
(iv),
(v),
that
the
data
(b
Θ
)
and
(c
Θ
)
act
on
proces-
sions
of
tensor
packets
that
arise
from
the
mono-analytic
étale-like
log-shells
constructed
from
the
data
(a
Θ
).
The
indeterminacies
(Ind1),(Ind2),
(Ind3)
referred
to
above
act
on
these
log-shells
and
may
be
described
as
follows:
(Ind1)
These
indeterminacies
are
the
étale-transport
indeterminacies
[cf.
§2.7,
(vi);
Example
2.12.3,
(i)]
that
occur
as
a
result
of
the
automorphisms
[which,
as
was
discussed
in
Example
2.12.3,
(i),
do
not,
in
general,
preserve
the
ring
structure!]
of
the
various
“G
k
’s”
that
appear
in
the
data
(a
Θ
).
(Ind2)
These
indeterminacies
are
the
Kummer-detachment
indeterminacies
[cf.
§2.7,
(vi)]
that
occur
as
a
result
of
the
identification
of,
or
confusion
between,
mono-
analytic
Frobenius-like
and
mono-analytic
étale-like
log-shells
[cf.
the
discussion
of
§3.6,
(iv)].
At
a
more
concrete
level,
these
indeterminacies
arise
from
the
action
of
the
group
of
“isometries”
—
which
is
often
denoted
“Ism(−)”
[cf.
[IUTchII],
Example
1.8,
(iv)]
—
of
the
data
“O
k
×μ
”
[which
we
regard
as
equipped
with
a
system
of
integral
structures,
or
lattices]
of
(a
Θ
),
i.e.,
the
compact
topological
group
of
G
k
-equivariant
automorphisms
of
the
ind-topological
module
O
k
×μ
that,
for
⊆
(O
k
×μ
)
H
.
each
open
subgroup
H
⊆
G
k
,
preserve
the
lattice
O
×μ
H
k
(Ind3)
These
indeterminacies
are
the
Kummer-detachment
indeterminacies
[cf.
§2.7,
(vi)]
that
occur
as
a
result
of
the
upper
semi-commutativity
[cf.
Example
2.12.3,
(iv);
§3.6,
(iv)]
of
the
log-Kummer
correspondence
[cf.
[IUTchIII],
Remark
3.12.2,
(iv),
(v)],
i.e.,
the
system
of
log-links
and
Kummer
isomorphisms
of
a
particular
vertical
line
of
the
log-theta-lattice
[cf.
Fig.
3.15].
In
addition
to
these
“explicitly
visible”
indeterminacies,
there
are
also
“invisible
in-
determinacies”
[cf.
[IUTchIII],
Remark
3.11.4]
that
in
fact
arise,
but
may
be
ignored
in
the
above
description,
essentially
as
a
formal
consequence
of
the
way
in
which
the
various
objects
that
appear
are
defined:
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
105
×
·
The
various
theta
values
and
copies
of
F
mod
that
occur
as
Frobenius-like
Galois
evaluation
output
data
at
various
vertical
positions
of
the
log-Kummer
correspondence
[cf.
the
discussion
of
§3.6,
(iv),
(v)]
satisfy
an
important
“non-
interference”
property
[cf.
[IUTchIII],
Proposition
3.5,
(ii),
(c);
[IUTchIII],
Proposition
3.10,
(ii)]:
namely,
the
intersection
of
such
output
data
with
the
product
of
the
local
units
[i.e.,
“O
×
”]
at
those
elements
of
V
at
which
the
out-
put
data
in
question
occurs
consists
only
of
roots
of
unity.
As
a
result,
the
only
“possible
confusion”,
or
“indeterminacy”,
that
occurs
as
a
consequence
of
possibly
applying
iterates
of
the
log-link
to
the
various
local
units
consists
of
a
possible
multiplication
by
a
root
of
unity.
On
the
other
hand,
since
the
theta
×
that
occur
as
Frobenius-like
Galois
evaluation
output
values
and
copies
of
F
mod
data
are
defined
in
such
a
way
as
to
be
stable
under
the
action
by
multiplica-
tion
by
such
roots
of
unity,
this
indeterminacy
may,
in
fact,
be
ignored
[cf.
the
discussion
of
[IUTchIII],
Remark
3.11.4,
(i)].
·
The
indeterminacy
of
possible
multiplication
by
±1
in
the
cyclotomic
rigidity
isomorphism
that
is
applied
in
the
Kummer
theory
of
κ-coric
ratio-
nal
functions
[cf.
the
final
portion
of
the
discussion
of
§3.4,
(ii)]
may
be
ignored
since
the
global
Frobenioids
related
to
the
data
(c
Θ
),
i.e.,
that
arise
from
copies
of
F
mod
,
only
require
the
use
of
the
totality
of
[copies
of]
the
multiplica-
×
,
which
is
stabilized
by
the
operation
of
inversion
[cf.
the
tive
monoid
F
mod
discussion
of
[IUTchIII],
Remark
3.11.4,
(i)].
At
this
point,
it
is
useful
to
recall
[cf.
the
discussion
at
the
beginning
of
§3.4,
(i)]
that
it
was
possible
to
define,
in
§3.3,
(vii),
the
gluing
isomorphisms
that
constitute
the
Θ-link
between
the
domain
data
(a
Θ
),
(b
Θ
),
(c
Θ
)
and
the
codomain
data
(a
q
),
(b
q
),
(c
q
)
precisely
because
we
worked
with
various
abstract
monoids
or
global
realified
Frobenioids,
i.e.,
as
opposed
to
the
“conventional
scheme-like
representations”
of
this
data
(a
Θ
),
(b
Θ
),
(c
Θ
)
in
terms
of
theta
values
and
copies
of
NF’s.
In
particular,
one
way
to
interpret
the
multiradial
representation
discussed
above
[cf.
the
discussion
of
“simultaneous
execution”
at
the
beginning
of
§2.9
and
§3.4,
(i);
the
discussion
of
the
properties
“IPL”,
“SHE”,
“APT”,
“HIS”
in
[IUTchIII],
Remark
3.11.1]
is
as
follows:
This
multiradial
representation
may
be
understood
as
the
[somewhat
sur-
prising!]
assertion
that
not
only
the
domain
data
(a
Θ
),
(b
Θ
),
(c
Θ
),
but
also
the
codomain
data
(a
q
),
(b
q
),
(c
q
)
—
or,
indeed,
any
collection
of
data
[i.e.,
not
just
the
codomain
data
(a
q
),
(b
q
),
(c
q
)!]
that
is
isomorphic
to
the
domain
data
(a
Θ
),
(b
Θ
),
(c
Θ
)
[cf.,
e.g.,
the
discussion
concerning
“q
λ
”
in
the
second
to
last
display
of
§3.8
below]
—
can,
when
regarded
up
to
suitable
indeterminacies,
be
represented
Shinichi
Mochizuki
106
{q
j
2
RHS
}
j
{q
j
..
.
{q
j
2
LHS
}
j
..
.
q
Θ
−→
“=”
q
RHS
2
LHS
}
j
!!
{q
j
2
RHS
}
j
mono-analytic
log-shells
LHS
Fig.
3.19:
The
gluing,
or
tautological
identification
“=”,
of
the
Θ-link
from
the
point
of
view
of
the
multiradial
representation
via
the
“conventional
scheme-like
representation”
of
(a
Θ
),
(b
Θ
),
(c
Θ
)
in
a
fashion
that
is
compatible
with
the
original
“conventional
scheme-like
representation”
of
the
given
collection
of
data
[i.e.,
such
as
the
“conventional
scheme-like
representation”
of
the
data
(a
q
),
(b
q
),
(c
q
)].
If
we
specialize
this
interpretation
to
the
case
of
the
data
(a
q
),
(b
q
),
(c
q
),
then
we
obtain
the
following
[again
somewhat
surprising!]
interpretation
of
the
multiradial
representa-
tion
discussed
above
[cf.
the
discussion
of
§2.4]:
If
one
takes
a
symmetrized
average
over
N
=
1
2
,
2
2
,
.
.
.
,
j
2
,
.
.
.
,
(l
)
2
,
and
one
works
up
to
suitable
indeterminacies,
then
the
arithmetic
line
bundle
determined
by
“q”
[i.e.,
the
2l-th
roots
of
the
q-parameter
at
the
valuations
∈
V
bad
,
or,
alterna-
tively,
the
“q-pilot
object”]
may
be
identified
—
i.e.,
from
the
point
of
view
of
performing
any
sort
of
computation
that
takes
into
account
the
“suitable
indeterminacies”
—
with
the
arithmetic
line
bundle
determined
by
“q
N
”
[i.e.,
the
“Θ-pilot
object”]
obtained
by
raising
the
arithmetic
line
bundle
deter-
mined
by
“q”
to
the
N
-th
tensor
power
[cf.
Fig.
3.19
above,
where
“LHS”
and
“RHS”
denote,
respectively,
the
left-hand
and
right-hand
sides,
i.e.,
the
domain
and
codomain,
of
the
Θ-link].
(ii)
Log-volume
estimates:
The
interpretation
discussed
in
the
final
portion
of
(i)
leads
naturally
to
an
estimate
of
the
arithmetic
degree
of
the
q-pilot
object
[cf.
[IUTchIII],
Theorem
B;
[IUTchIII],
Corollary
3.12],
as
follows
[cf.
Steps
(x),
(xi)
of
the
proof
of
[IUTchIII],
Corollary
3.12;
[IUTchIII],
Fig.
3.8]:
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
(1
est
(2
est
107
)
One
starts
with
the
Frobenius-like
version
of
the
q-pilot
object
—
i.e.,
“q
”
RHS
—
on
the
RHS
of
the
Θ-link.
All
subsequent
computations
are
to
be
understood
as
computations
that
are
performed
relative
to
the
fixed
arithmetic
holomorphic
structure
of
this
RHS
of
the
Θ-link.
)
The
isomorphism
class
determined
by
this
q-pilot
object
in
the
global
realified
Frobenioid
of
(c
q
)
[cf.
§3.3,
(vii)]
is
sent,
via
the
Θ-link,
to
the
isomorphism
class
2
determined
by
the
Θ-pilot
object
—
i.e.,
“{q
j
}
j
”
—
in
the
global
realified
Frobenioid
of
(c
Θ
)
[cf.
§3.3,
(vii)].
(3
est
(4
est
(5
est
(6
est
(7
est
(8
est
LHS
)
One
then
applies
the
multiradial
representation
discussed
in
(i)
[cf.
Fig.
3.19].
)
One
observes
that
the
log-volume,
suitably
normalized,
on
the
log-shells
that
occur
in
this
multiradial
representation
is
invariant
with
respect
to
the
indetermi-
nacies
(Ind1)
and
(Ind2),
as
well
as
with
respect
to
the
invisible
indeterminacies,
discussed
in
(i).
)
On
the
other
hand,
the
upper
semi-commutativity
indeterminacy
(Ind3)
—
i.e.,
“commutativity”
at
the
level
of
inclusions
of
regions
in
log-shells
[cf.
the
discus-
sion
of
Example
2.12.3,
(iv)]
—
may
be
understood
as
asserting
that
the
log-volume
of
the
multiradial
representation
of
the
Θ-pilot
object
must
be
interpreted
as
an
upper
bound.
2
)
The
multiradial
representation
of
the
Θ-pilot
object
“{q
j
}
j
”
can
only
be
LHS
compared
to
the
isomorphism
class
determined
by
the
original
q-pilot
object
”
in
the
global
realified
Frobenioid
of
(c
q
),
i.e.,
not
to
the
specific
arithmetic
“q
RHS
line
bundle
given
by
a
copy
of
the
trivial
arithmetic
line
bundle
with
fixed
local
trivializations
“1”
multiplied
by
2l-th
roots
of
q-parameters
[cf.
[IUTchIII],
Remarks
3.9.5,
(vii),
(viii),
(ix),
(x);
Remark
3.12.2,
(v)].
)
In
particular,
in
order
to
perform
such
a
comparison
between
the
multiradial
2
representation
of
the
Θ-pilot
object
“{q
j
}
j
”
and
the
isomorphism
class
de-
LHS
”
in
the
global
realified
Frobenioid
of
termined
by
the
original
q-pilot
object
“q
RHS
q
(c
),
it
is
necessary
to
make
the
output
data
of
the
multiradial
representation
into
a
collection
of
“O
k
-modules”
[where
we
use
the
notation
“O
k
”
to
denote
the
various
completions
of
the
ring
of
integers
of
K
at,
for
simplicity,
the
valuations
∈
V
non
],
i.e.,
relative
to
the
arithmetic
holomorphic
structure
of
the
RHS
of
the
Θ-link!
)
Such
“O
k
-modules”
are
obtained
by,
essentially
[cf.
[IUTchIII],
Remark
3.9.5,
for
more
details],
forming
the
“O
k
-modules
generated
by”
the
various
tensor
packets
of
log-shells
[cf.
the
discussion
of
§3.6,
(iv),
(v)]
that
appear
in
the
multiradial
Shinichi
Mochizuki
108
representation,
i.e.,
which,
a
priori,
are
[up
to
a
factor
given
by
a
suitable
power
of
“p”]
just
topological
modules
“log(O
k
×
)
⊗
log(O
k
×
)
⊗
.
.
.
⊗
log(O
k
×
)”
—
that
is
to
say,
tensor
products
of
j
+
1
copies
of
“log(O
k
×
)”
at
the
portion
of
the
multiradial
representation
labeled
by
j.
This
operation
yields
a
slightly
enlarge-
ment
of
the
multiradial
representation,
which
is
referred
to
as
the
holomorphic
hull
[cf.
[IUTchIII],
Corollary
3.12;
[IUTchIII],
Remark
3.9.5]
of
the
multiradial
representation.
(9
est
)
The
log-volume,
when
applied
to
the
original
q-pilot
object
“q
2
RHS
”
or
to
the
Θ-pilot
object
“{q
j
}
j
”,
may
be
interpreted
as
the
arithmetic
degree
of
these
LHS
objects
[cf.
§2.2;
[IUTchIII],
Remark
1.5.2,
(i),
(iii);
[IUTchIII],
Proposition
3.9,
(iii);
[IUTchIII],
Remark
3.10.1,
(iii)].
(10
est
)
In
particular,
any
upper
bound
on
the
log-volume
of
the
holomorphic
hull
of
the
2
multiradial
representation
of
the
Θ-pilot
object
“{q
j
}
j
”
may
be
interpreted
LHS
[cf.
the
discussion
of
[IUTchIII],
Remark
3.9.5,
(vii),
(viii),
(ix);
[IUTchIII],
Remark
3.11.1;
[IUTchIII],
Remark
3.12.2,
(i),
(ii)]
as
an
upper
bound
on
the
log-volume
of
the
original
[Frobenius-like]
q-pilot
object
“q
”.
RHS
(11
est
(12
est
)
This
comparison
of
log-volumes
was
obtained
by
considering
the
images
of
var-
ious
Frobenius-like
objects
in
the
étale-like
tensor
packets
of
log-shells
of
the
multi-
radial
representation.
In
particular,
one
must
apply
the
log-Kummer
correspon-
2
dence
on
both
the
LHS
[in
the
case
of
the
Θ-pilot
object
“{q
j
}
j
”]
and
the
RHS
LHS
”]
of
the
Θ-link.
On
[in
the
case
of
the
original
[Frobenius-like]
q-pilot
object
“q
RHS
the
other
hand,
this
does
not
affect
the
resulting
inequality,
in
light
of
the
com-
patibility
of
log-volumes
with
the
arrows
of
the
log-Kummer
correspondence
[cf.
Example
2.12.3,
(iv);
§3.6,
(iv),
as
well
as
the
discussion
of
[IUTchIII],
Remark
3.9.5,
(vii),
(viii),
(ix);
[IUTchIII],
Remark
3.12.2,
(iv),
(v)].
)
Thus,
in
summary,
one
obtains
an
inequality
of
log-volumes
[cf.
[IUTchIII],
Theorem
B;
[IUTchIII],
Corollary
3.12]
⎛
⎞
Θ-pilot
object
up
to
⎛
⎞
⎜
mild
indeterminacies,
⎟
⎜
⎟
⎜
⎟
⎜
⎟
log-vol.
⎜
i.e.,
(Ind1),
(Ind2),
(Ind3),
⎟
≥
log-vol.
⎝
q-pilot
object
⎠
(≈
0)
⎜
⎟
⎠
⎝
plus
formation
of
holomorphic
hull
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
109
—
where
the
log-volume
of
the
q-pilot
object
on
the
right-hand
side
of
the
inequality
is
negative
and
of
negligible
absolute
value
by
comparison
to
the
terms
of
interest
[to
be
discussed
in
more
detail
in
(iv)
below]
on
the
left-hand
side
of
the
inequality.
(iii)
Comparison
with
a
result
of
Stewart-Yu:
Recall
that
in
[StYu],
an
in-
equality
is
obtained
which
may
be
thought
of
as
a
sort
of
“weak
version
of
the
ABC
Conjecture”,
i.e.,
which
is,
roughly
speaking,
weaker
than
the
inequality
of
the
usual
ABC
Conjecture
in
that
it
contains
an
undesired
exponential
operation
“exp(−)”
in
its
upper
bound.
This
sort
of
deviation
from
the
inequality
of
the
usual
ABC
Conjec-
ture
is
of
interest
from
the
point
of
view
of
the
“vertical
shift”
discussed
in
§3.6,
(iv),
which,
on
the
one
hand,
gives
rise
to
the
indeterminacy
(Ind3)
[cf.
the
discussion
est
of
(i);
(ii),
(5
)]
and,
on
the
other
hand,
arises
from
the
fact
that
the
horizontally
coric
portion
of
the
data
related
by
the
Θ-link
differs
from
the
sort
of
data
in
which
one
is
ultimately
interested
precisely
by
a
single
iterate
of
the
log-link,
i.e.,
a
single
vertical
shift
in
the
log-theta-lattice.
(iv)
Computation
of
log-volumes:
Let
us
return
to
the
discussion
of
(ii).
It
est
remains
to
compute
the
left-hand
side
of
the
inequality
of
(ii),
(12
),
in
more
elemen-
tary
terms.
This
is
done
in
[IUTchIV],
Theorem
1.10.
This
computation
yields
a
rather
strong
version
of
the
Szpiro
Conjecture
inequality,
in
the
case
of
elliptic
curves
over
NF’s
that
admit
initial
Θ-data
[cf.
§3.3,
(i)]
that
satisfies
certain
technical
condi-
tions.
The
existence
of
such
initial
Θ-data
that
satisfies
certain
technical
conditions
is
then
verified
in
[IUTchIV],
Corollary
2.2,
(ii),
for
elliptic
curves
over
NF’s
that
satisfy
certain
technical
conditions
by
applying
the
techniques
of
[GenEll],
§3,
§4.
Here,
we
remark
in
passing
that
the
prime
number
l
that
appears
in
this
initial
Θ-data
con-
structed
in
[IUTchIV],
Corollary
2.2,
(ii),
is
roughly
of
the
order
of
the
square
root
of
the
height
of
the
elliptic
curve
under
consideration
[cf.
[IUTchIV],
Corollary
2.2,
(ii),
(C1)].
This
initial
Θ-data
yields
a
version
of
the
Szpiro
Conjecture
inequality
[cf.
[IUTchIV],
Corollary
2.2,
(ii),
(iii)],
which,
although
somewhat
weaker
and
less
effective
than
the
inequality
of
[IUTchIV],
Theorem
1.10,
is
still
rather
strong
in
the
sense
that
it
implies
that,
if
we
restrict,
for
simplicity,
to
the
case
of
elliptic
curves
over
Q,
then
the
“
terms”
that
appear
in
the
Szpiro
Conjecture
inequality
concerning
the
height
h
may
be
bounded
above
by
terms
of
the
order
of
h
1/2
·
log(h)
—
i.e.,
at
least
in
the
case
of
elliptic
curves
over
Q
whose
moduli
are
“com-
pactly
bounded”,
in
the
sense
that
the
moduli
lie
inside
given
fixed
compact
subsets
of
the
sets
of
rational
points
of
the
moduli
of
elliptic
curves
over
R
and
Q
2
110
Shinichi
Mochizuki
[cf.
[IUTchIV],
Remark
1.10.5,
(ii),
(iii);
[IUTchIV],
Remark
2.2.1,
(i),
(ii)].
Here,
we
recall
from
these
Remarks
in
[IUTchIV]
[cf.
also
[Mss];
[vFr],
§2]
that
This
“
12
”
in
the
exponent
of
h
is
of
interest
in
light
of
the
existence
of
sequences
of
“abc
sums”
for
which
this
“
12
”
is
asympototically
attained,
i.e.,
as
a
bound
from
below,
but
only
if
one
works
with
abc
sums
that
correspond
to
elliptic
curves
whose
moduli
are
not
necessarily
compactly
bounded.
This
prompts
the
following
question:
Can
one
construct
sequences
of
abc
sums
with
similar
asymptotic
behavior,
but
which
correspond
to
elliptic
curves
whose
moduli
are
indeed
compactly
bounded?
At
the
time
of
writing,
it
appears
that
no
definitive
answer
to
this
question
is
known,
although
there
does
exist
some
preliminary
work
in
this
direction
[cf.
[Wada]].
In
this
context,
it
is
also
of
interest
to
recall
[cf.
the
discussion
of
[IUTchIV],
Remark
2.2.1;
[vFr],
§2]
that
this
“
12
”
is
highly
reminiscent
of
the
“
12
”
that
appears
in
the
Riemann
hypothesis.
So
far,
in
the
above
discussion,
we
have
restricted
ourselves
to
versions
of
the
Szpiro
Conjecture
inequality
for
elliptic
curves
over
NF’s
that
satisfy
various
technical
conditions.
On
the
other
hand,
by
applying
the
theory
of
noncritical
Belyi
maps
[cf.
the
discussion
in
the
fi-
nal
portion
of
§2.1;
[GenEll],
Theorem
2.1;
[IUTchIV],
Corollary
2.3;
[IUTchIV],
Theorem
A]
—
which
may
be
thought
of
as
a
sort
of
arithmetic
version
of
analytic
continuation
[cf.
the
discussion
of
§3.3,
(vi);
[IUTchI],
Remark
5.1.4;
[IUTchIV],
Remark
2.2.4,
(iii)]
—
one
may
derive
the
inequalities
of
the
Vojta/Szpiro/ABC
Conjectures
in
their
usual
form.
We
refer
to
[Fsk],
§1.3,
§2.12,
for
another
—
and,
in
certain
respects,
more
detailed
—
discussion
of
these
aspects
of
inter-universal
Teichmüller
theory.
Although
a
detailed
discussion
of
the
somewhat
technical,
but
entirely
elementary
computation
of
the
left-
est
hand
side
of
the
inequality
of
(ii),
(12
),
lies
beyond
the
scope
of
the
present
paper,
we
conclude
the
present
(iv)
with
a
summary
of
the
very
simple
computation
of
the
est
leading
term
of
the
left-hand
side
of
the
inequality
of
(ii),
(12
):
·
First,
one
notes
[cf.
[IUTchIV],
Proposition
1.2,
(i);
[IUTchIV],
Proposition
1.3,
(i);
the
second
to
last
display
of
Step
(v)
of
the
proof
of
[IUTchIV],
Theorem
1.10]
that,
if
one
ignores
[since
we
are
only
interested
in
computing
the
leading
term!]
the
archimedean
valuations
of
K,
as
well
as
the
nonarchimedean
valua-
tions
of
K
whose
ramification
index
over
Q
is
“large”
[i.e.,
is
≥
the
cardinality
of
the
set
of
nonzero
elements
of
the
residue
field
of
the
corresponding
prime
of
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
111
Q],
then
the
resulting
log-volume
[suitably
normalized!]
of
the
tensor
packet
of
log-shells
corresponding
to
the
label
j
∈
{1,
2,
.
.
.
,
l
}
is
(j
+
1)
·
log(d
K
)
—
where
we
write
log(d
K
)
for
the
arithmetic
degree
[suitably
normalized,
so
as
to
be
invariant
with
respect
to
finite
extensions
—
cf.
[IUTchIV],
Definition
1.9,
(i)]
of
the
arithmetic
divisor
determined
by
the
different
ideal
of
the
number
field
K
over
Q.
·
On
the
other
hand,
the
effect
—
i.e.,
on
the
tensor
packet
of
log-shells
corre-
2
sponding
to
the
label
j
—
of
multiplying
by
the
theta
value
“q
j
”
[cf.
Figs.
3.16,
3.18]
at
v
∈
V
bad
[cf.
the
second
to
last
display
of
Step
(v)
of
the
proof
of
[IUTchIV],
Theorem
1.10]
is
given
by
2
−
j
2l
·
log(q)
—
where
we
write
log(q)
for
the
arithmetic
degree
[again
suitably
normalized]
of
the
arithmetic
divisor
determined
by
the
q-parameters
of
the
elliptic
curve
E
over
F
at
the
valuations
that
lie
over
valuations
∈
V
bad
mod
.
·
Thus,
the
leading
term
of
the
log-volume
of
the
left-hand
side
of
the
est
inequality
of
(ii),
(12
),
is
given
by
l
2
(j
+
1)
·
log(d
K
)
−
j
2l
·
log(q)
≈
K
1
l
2
2
·
(
2
)
·
log(d
)
1
−
3·2l
·
(
2
l
)
3
·
log(q)
j=1
=
K
l
2
8
·
log(d
)
=
K
l
2
48
{6
·
log(d
)
2
l
−
48
·
log(q)
−
log(q)}
—
where
the
notation
“≈”
denotes
a
possible
omission
of
terms
that
do
not
affect
the
leading
term.
That
is
to
say,
one
obtains
a
“large
positive
constant”
l
2
48
times
precisely
the
quantity
—
i.e.,
6
·
log(d
K
)
−
log(q)
—
that
one
wishes
to
bound
from
below
in
order
to
conclude
[a
suitable
version
of]
the
Szpiro
Conjecture
inequality.
As
discussed
in
[IUTchIV],
Remark
1.10.1
[cf.
also
the
discussion
of
§3.9,
(i),
(ii),
below],
this
“computation
of
the
leading
term”,
which
was
originally
motivated
by
the
scheme-
theoretic
Hodge-Arakelov
theory
of
[HASurI],
[HASurII],
was
in
fact
known
to
the
author
around
the
year
2000.
Put
another
way,
one
of
the
primary
motivations
for
the
development
of
inter-universal
Teichmüller
theory
was
precisely
the
problem
of
establishing
a
suitable
framework,
or
geometry,
in
which
this
computation
could
be
performed.
Shinichi
Mochizuki
112
§
3.8.
Comparison
with
the
Gaussian
integral
At
this
point,
it
is
of
interest
to
compare
and
contrast
the
theory
of
“arithmetic
changes
of
coordinates”
[i.e.,
§2]
and
multiradial
representations
[i.e.,
the
present
§3]
discussed
so
far
in
the
present
paper
with
the
classical
computation
of
the
Gaussian
integral,
as
discussed
in
§1.
In
the
following,
the
various
“Steps”
refer
to
the
“Steps”
in
the
computation
of
the
Gaussian
integral,
as
reviewed
in
§1.
(1
)
The
naive
change
of
coordinates
“e
−x
u”
of
Step
1
[cf.
also
Step
2]
is
formally
reminiscent
[cf.
[IUTchII],
Remark
1.12.5,
(ii)]
of
the
assignment
2
gau
2
{q
j
}
j=1,...,l
→
q
that
appears
in
the
definition
of
the
Θ-link
[cf.
§2.4;
§3.3,
(vii),
(b
Θ
),
(b
q
)].
(2
gau
(3
gau
(4
gau
(5
gau
)
The
introduction
of
two
“mutually
alien”
copies
of
the
Gaussian
integral
in
Step
3
may
be
thought
of
as
corresponding
to
the
appearance
of
the
two
Θ
±ell
N
F
-
Hodge
theaters
“•”
in
the
domain
and
codomain
of
the
Θ-link
[cf.
§2.7,
(i);
the
two
vertical
lines
in
the
right-hand
portion
of
Fig.
3.6],
which
may
be
thought
of
as
representing
two
“mutually
alien”
copies
of
the
conventional
scheme
theory
sur-
rounding
the
elliptic
curve
under
consideration,
i.e.,
the
elliptic
curve
that
appears
in
the
initial
Θ-data
of
§3.3,
(i).
)
The
two
dimensions
of
the
Euclidean
space
R
2
that
appears
in
Step
4
may
be
thought
of
as
corresponding
to
the
two
dimensions
of
the
log-theta-lattice
[cf.
§3.3,
(ii)],
which
are
closely
related
to
the
two
underlying
combinatorial
dimensions
of
a
ring.
Here,
we
recall
from
§2.11
that
these
two
underlying
com-
binatorial
dimensions
of
a
ring
may
be
understood
quite
explicitly
in
the
case
of
NF’s,
MLF’s,
or
the
field
of
complex
numbers.
)
The
point
of
view
discussed
in
Step
5
that
integrals
may
be
thought
of
as
com-
putations
of
net
masses
as
limits
of
sums
of
infinitesimals
of
zero
mass
may
be
understood
as
consisting
of
two
aspects:
First
of
all,
the
summation
of
local
contri-
butions
that
occurs
in
an
integral
may
be
regarded
as
corresponding
to
the
use
of
prime-strips
[i.e.,
local
data
indexed
by
elements
of
V
—
cf.
the
discussion
of
§3.3,
(iv)]
and
the
computation
of
heights
in
terms
of
log-volumes,
as
discussed
in
§2.2.
On
the
other
hand,
the
limit
aspect
of
an
integral,
which
involves
the
consideration
of
some
sort
of
notion
of
infinitesimals
[i.e.,
“zero
mass”
objects],
may
be
thought
of
as
corresponding
to
the
fundamental
dichotomy
between
Frobenius-like
and
étale-like
objects
[cf.
§2.2;
§2.7,
(ii),
(iii);
§2.8;
§2.9;
§3.3,
(iii);
§3.4,
(i);
§3.5].
)
The
generalities
concerning
the
effect
on
integrals
of
changes
of
coordinates
on
local
patches
of
the
Euclidean
plane
in
Step
6
may
be
thought
of
as
corresponding
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
Classical
computation
of
the
Gaussian
integral
Inter-universal
Teichmüller
theory
Θ-link
naive
change
of
coordinates
2
e
−x
u
{q
}
j=1,...,l
→
q
two
“mutually
alien”
copies
of
the
Gaussian
integral
domain
and
codomain
“•”
of
the
Θ-link
two
dimensions
of
the
Euclidean
plane
R
2
two
dimensions
of
the
log-theta-lattice/
rings/NF’s/MLF’s/C
net
mass
=
(infinitesimals
of
zero
mass)
prime-strips;
heights
as
log-volumes;
Frobenius-like/étale-like
objects
lim
j
2
the
effect
on
integrals
of
local
Euclidean
changes
of
coordinates
computation
via
mono-anabelian
transport
of
the
effect
of
arithmetic
changes
of
coordinates
unipotent
linear
transformations,
toral
dilations,
and
rotations
examples
for
MLF’s
of
arithmetic
changes
of
coordinates
Fig.
3.20.1:
Comparison
between
inter-universal
Teichmüller
theory
and
the
classical
computation
of
the
Gaussian
integral
113
Shinichi
Mochizuki
114
to
the
generalities
on
the
computational
technique
of
mono-anabelian
transport
as
a
mechanism
for
computing
the
effect
of
“arithmetic
changes
of
coordinates”
[cf.
the
discussion
of
§2.7,
§2.8,
§2.9,
§2.10,
§2.11].
This
technique
is
motivated
by
the
concrete
examples
given
in
§2.5,
§2.6
of
changes
of
coordinates
related
to
positive
characteristic
Frobenius
morphisms,
as
well
as
by
the
discussion
of
examples
of
finite
discrete
approximations
of
harmonic
analysis
given
in
§2.14.
(6
gau
(7
gau
(8
gau
(9
gau
)
The
fundamental
examples
given
in
Step
7
of
linear
changes
of
coordinates
in
the
Euclidean
plane
—
i.e.,
unipotent
linear
transformations,
toral
dilations,
and
rotations
—
may
be
thought
of
as
corresponding
to
the
fundamental
examples
of
arithmetic
changes
of
coordinates
in
the
case
of
MLF’s
that
were
discussed
in
§2.12
[cf.
also
the
discussion
of
§2.3,
§2.4,
and
§2.13;
§3.3,
(vi)].
)
The
passage
from
planar
cartesian
to
polar
coordinates
discussed
in
Step
8
may
be
understood
as
a
sort
of
rotation,
or
continuous
deformation,
between
the
two
mutually
alien
copies
of
the
Gaussian
integral
introduced
in
Step
3,
i.e.,
which
correspond
to
the
x-
and
y-axes.
Thus,
this
passage
to
polar
coordinates
may
be
regarded
as
corresponding
[cf.
the
discussion
at
the
beginning
of
§3.1;
§3.1,
(iv);
§3.2;
the
discussion
surrounding
Figs.
3.1,
3.3,
3.5,
3.6,
3.12,
3.13,
3.19;
the
discus-
sion
of
[IUTchII],
Remark
1.12.5]
to
the
“deformation”,
or
“parallel
transport”,
between
distinct
collections
of
radial
data
that
appears
in
the
definition
of
the
notion
of
multiradiality
and,
in
particular,
to
the
passage
from
the
log-theta-
lattice
to
the
étale-picture
[cf.
§3.6,
(i)]
and,
ultimately,
to
the
multiradial
representation
of
§3.7,
(i).
)
The
fundamental
decoupling
into
radial
and
angular
coordinates
discussed
in
Step
9
may
be
understood
as
corresponding
to
the
discussion
in
§3.4
of
Kummer
theory
for
special
types
of
functions
via
multiradial
decouplings/cyclotomic
rigidity
[cf.
also
the
discussion
of
unit
group
and
value
group
portions
in
§2.11;
§3.3,
(vii)].
)
The
efficacy
of
the
change
of
coordinates
that
renders
possible
the
evaluation
of
the
radial
integral
in
Step
10
may
be
understood
as
an
essentially
formal
con-
sequence
of
the
quadratic
nature
of
the
exponent
that
appears
in
the
Gaussian
distribution.
This
fundamental
aspect
of
the
computation
of
the
Gaussian
integral
may
be
regarded
as
corresponding
to
the
fact
that
the
rigidity
properties
of
mono-
theta
environments
that
underlie
the
multiradial
decouplings/cyclotomic
rigidity
discussed
in
§3.4,
(iii),
(iv)
are,
in
essence,
formal
consequences
[cf.
the
discussion
in
the
final
portion
of
§3.4,
(iv)]
of
the
quadratic
structure
of
the
com-
mutators
of
the
theta
groups
associated
to
the
ample
line
bundles
that
appear
in
the
theory
[cf.
the
discussion
of
[IUTchIII],
Remark
2.1.1;
[IUTchIV],
Remark
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
Classical
computation
of
the
Gaussian
integral
Inter-universal
Teichmüller
theory
passage
from
planar
cartesian
to
polar
coordinates
passage
from
the
log-theta-lattice
to
the
étale-picture;
multiradiality
decoupling
into
radial
and
angular
coordinates
Kummer
theory
for
functions
via
multiradial
decouplings/
cyclotomic
rigidity
evaluation
of
the
radial
integral
via
the
quadraticity
of
the
exponent
in
the
Gaussian
distribution
multiradiality/
mono-theta
rigidity
via
the
quadraticity
of
theta
group
commutators;
value
group
portion
of
the
Θ-link
evaluation
of
the
angular
integral
via
the
natural
logarithm
on
the
complex
units
log-shells
as
containers
for
Galois
evaluation
output;
iterated
rotations/juggling
of
the
log-link
acting
on
[local]
units
naive
change
of
coordinates
“justifiable”
up
to
a
suitable
√
“error
factor”
π
arising
from
the
square
root
of
the
angular
integral
over
the
complex
units
naive
approach
to
bounding
heights
via
“Gaussian
Frobenius
morphisms”
on
NF’s
“justifiable”
up
to
a
suitable
“log-different
error
factor”
arising
from
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
acting
on
log-shells
Fig.
3.20.2:
Comparison
between
inter-universal
Teichmüller
theory
and
the
classical
computation
of
the
Gaussian
integral
115
Shinichi
Mochizuki
116
2.2.2;
the
discussion
of
the
functional
equation
of
the
theta
function
in
[Pano],
§3].
In
particular,
the
evaluation
of
the
radial
integral
in
Step
10
corresponds
to
the
por-
tion
of
inter-universal
Teichmüller
theory
that
relates
to
the
[local]
value
group
gau
portion
(b
Θ
)
of
the
Θ-link
[cf.
(1
)].
(10
gau
(11
gau
)
The
angular
integral
of
Step
11
is
an
integral
over
the
unit
group
of
the
field
of
complex
numbers
that
is
evaluated
by
executing
the
change
of
coordinates
determined
by
the
imaginary
part
of
the
natural
logarithm.
This
important
aspect
of
the
computation
of
the
Gaussian
integral
may
be
regarded
as
corresponding
to
the
theory
of
Galois
evaluation
and
log-shells
exposed
in
§3.6
—
cf.,
especially,
the
theory
involving
iterates
of
the
log-link
discussed
in
§3.6,
(iv);
§3.7,
(i).
In
this
context,
it
is
of
interest
to
recall
[cf.
Example
2.12.3,
(v);
§3.3,
(ii),
(vi)]
that
the
log-link
may
be
understood
as
a
sort
of
arithmetic
rotation,
or
juggling,
of
the
two
underlying
combinatorial
dimensions
of
a
ring
that
essentially
concerns
the
[local]
unit
group
portion,
i.e.,
(a
Θ
),
(a
q
),
of
the
Θ-link
[cf.
§3.3,
(vii);
§3.6,
(iv)].
)
The
final
computation
of
the
Gaussian
integral
in
Step
12
may
be
summarized
gau
[cf.
§1.7]
as
asserting
that
the
naive
change
of
coordinates
of
(1
)
may
in
fact
be
“justified”,
provided
that
one
allows
for
a
suitable
“error
factor”
given
by
the
gau
square
root
of
the
angular
integral
of
(10
).
This
conclusion
may
be
understood
as
corresponding
to
the
computation,
discussed
in
§3.7,
(ii),
(iii),
(iv),
of
the
est
left-hand
side
of
the
inequality
of
§3.7,
(ii),
(12
).
This
computation
may
be
summarized
as
asserting
that
one
obtains
a
bound
on
the
height
of
the
elliptic
curve
under
consideration
whose
leading
term
is
the
“log-different
log(d
K
)”
[cf.
the
final
portion
of
§3.7,
(iv)]
that
one
expects
from
[a
suitable
version
of]
the
Szpiro
Conjecture.
Put
another
way,
this
computation
may
be
summarized
as
asserting
that
the
naive
approach
outlined
in
§2.3,
§2.4
to
bounding
heights
via
“Gaussian
Frobenius
morphisms”
on
NF’s
may
in
fact
be
“justified”,
provided
that
one
allows
for
a
suitable
“error
factor”
that
arises
from
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
acting
on
the
log-shells
—
i.e.,
the
[local]
unit
group
portion
(a
Θ
),
(a
q
)
of
the
Θ-link
[cf.
the
gau
discussion
of
(10
)]
—
that
appear
in
the
multiradial
representation
discussed
in
§3.7,
(i)
[cf.
also
the
discussion
of
[IUTchIV],
Remark
2.2.2].
Finally,
in
this
context,
we
observe
[cf.
the
final
portion
of
Step
12]
that,
just
as
in
the
case
of
the
computation
of
the
Gaussian
integral,
it
is
essentially
a
hopeless
task
2
∞
to
identify
“explicit
portions”
of
the
original
Gaussian
integral
−∞
e
−x
dx
on
the
real
line
that
“correspond”
precisely,
in
some
sort
of
meaningful
sense,
to
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
117
the
radial
and
angular
integrals
of
Steps
10
and
11,
it
is
essentially
a
hopeless
task
to
trace,
in
some
sort
of
explicit
or
readily
computable
fashion,
the
way
in
which
the
value
group
portions
(b
Θ
),
(c
Θ
)
in
the
domain
of
the
gau
Θ-link
[cf.
(9
)]
appear
within
the
multiradial
representation
via
tensor
gau
packets
of
log-shells
[cf.
(10
)]
expressed
in
objects
arising
from
the
codomain
of
the
Θ-link
[cf.
the
discussion
of
“APT”
in
[IUTchIII],
Remark
3.11.1,
(iv)].
The
various
observations
discussed
in
the
present
§3.8
are
summarized
in
Figs.
3.20.1,
gau
3.20.2,
above.
Finally,
with
regard
to
(1
),
we
note
that
2
the
left-hand
side
“{q
j
}
j=1,...,l
”
of
the
assignment
discussed
in
(1
not
be
replaced
by
gau
)
can-
“q
λ
”
for
1
=
λ
∈
Q
>0
or
by
“{q
N
·j
}
j=1,...,l
”
for
2
≤
N
∈
N.
2
Indeed,
this
property
of
the
left-hand
side
of
the
assignment
discussed
in
(1
case
of
“q
λ
”,
a
consequence
of
the
gau
)
is,
in
the
·
the
lack
[i.e.,
in
the
case
of
“q
λ
”]
of
a
theory
of
multiradial
decouplings/cyc-
lotomic
rigidity
of
the
sort
[cf.
the
discussion
of
§3.4,
(iii),
(iv)]
that
exists
in
the
case
of
theta
functions
and
mono-theta
environments
[cf.
the
discussion
of
[IUTchIII],
Remark
2.2.2,
(i),
(ii),
(iii)]
and,
in
the
case
of
2
“{q
N
·j
}
j=1,...,l
”,
a
consequence
of
the
·
special
role
[cf.
the
discussion
of
§3.4,
(iii)]
played
by
the
first
power
of
[reciprocals
of
l-th
roots
of
the]
theta
function;
·
the
condition
[cf.
the
discussion
at
the
beginning
of
§3.6]
that
the
assignment
“abstract
functions
→
values”
that
occurs
in
the
passage
from
theta
functions
to
theta
values
be
obtained
by
applying
the
technique
of
Galois
evaluation
[cf.
[IUTchII],
Remark
3.6.4,
(iii),
(iv);
[IUTchIII],
Remark
2.1.1,
(iv);
the
discussion
of
the
final
portion
of
Step
(xi)
of
the
proof
of
[IUTchIII],
Corollary
3.12].
Moreover,
gau
the
negation
of
this
property
of
the
left-hand
side
of
the
assignment
discussed
in
(1
)
would
imply
a
stronger
version
of
the
Szpiro
Conjecture
inequality
that
is
in
fact
false
[cf.
[IUTchIV],
Remark
2.3.2,
(ii)].
By
contrast,
Shinichi
Mochizuki
118
the
right-hand
side
“q”
of
the
assignment
discussed
in
(1
gau
)
can
be
replaced
by
“q
”
for
1
=
λ
∈
Q
>0
,
without
any
substantive
effect
on
the
theory;
moreover,
doing
so
does
not
result
in
any
substantive
improvement
in
the
estimates
discussed
in
§3.7,
(ii),
(iii),
(iv)
λ
[cf.
[IUTchIII],
Remark
3.12.1,
(ii)].
In
this
context,
it
is
of
interest
to
observe
that:
This
sort
of
qualitative
difference
between
the
left-
and
right-
hand
sides
of
the
2
assignment
{q
j
}
j=1,...,l
→
q
is
reminiscent
of
the
qualitative
difference
—
e.g.,
the
presence
or
absence
of
the
exponential!
—
between
the
left-
and
2
right-
hand
sides
of
the
naive
change
of
coordinates
e
−x
u.
§
3.9.
Relation
to
scheme-theoretic
Hodge-Arakelov
theory
In
the
present
§3.9,
we
pause
to
reconsider
the
theory
of
multiradial
represen-
tations
developed
in
the
present
§3
from
the
point
of
view
of
the
scheme-theoretic
Hodge-Arakelov
theory
discussed
in
Example
2.14.3
—
a
theory
which,
as
discussed
in
the
final
portion
of
§3.7,
(iv),
played
a
central
role
in
motivating
the
development
of
inter-universal
Teichmüller
theory.
(i)
Hodge
filtrations
and
theta
trivializations:
We
begin
by
examining
the
natural
isomorphism
of
F
-vector
spaces
of
dimension
l
2
Γ(E
†
,
L|
E
†
)
<l
∼
→
L|
E[l]
that
constitutes
the
fundamental
theorem
of
Hodge-Arakelov
theory
discussed
in
Example
2.14.3
in
a
bit
more
detail
[cf.,
e.g.,
the
discussion
surrounding
[Pano],
Theorem
1.1]
in
the
case
where
F
is
an
NF.
First
of
all,
we
observe
that,
although
both
the
domain
and
codomain
of
this
isomorphism
are
F
-vector
spaces
of
dimension
l
2
,
by
considering
the
natural
action
of
suitable
theta
groups
on
the
domain
and
codomain
and
applying
the
well-known
theory
of
irreducible
representations
of
theta
groups,
one
may
conclude
that,
up
to
“uninteresting
redudancies”,
this
isomorphism
may
in
fact
be
[“essentially”]
regarded
as
an
isomorphism
between
F
-vector
spaces
of
dimension
l.
The
left-hand
side
of
this
isomorphism
of
l-dimensional
F
-vector
spaces
admits
a
natural
Hodge
filtration
that
arises
by
considering
the
subspaces
of
relative
degree
<
t,
for
t
=
0,
.
.
.
,
l
−
1.
Moreover,
one
verifies
easily
that,
if
we
write
ω
E
for
the
cotangent
space
of
E
at
the
origin
of
E,
and
τ
E
for
the
dual
of
ω
E
,
then
the
adjacent
subquotients
of
this
Hodge
filtration
are
the
1-dimensional
F
-vector
spaces
τ
E
⊗t
for
t
=
0,
.
.
.
,
l
−
1,
tensored
with
some
fixed
1-dimensional
F
-vector
space
[i.e.,
which
is
independent
of
t],
which
we
shall
ignore
since
its
arithmetic
degree
[i.e.,
when
regarded
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
119
as
being
equipped
with
natural
integral
structures
at
the
nonarchimedean
valuations
of
F
and
natural
Hermitian
structures
at
the
archimedean
valuations
of
F
]
is
sufficiently
small
that
its
omission
does
not
affect
the
computation
of
the
leading
terms
of
interest.
On
the
other
hand,
the
right-hand
side
of
the
isomorphism
under
consideration
admits
a
natural
theta
trivialization
[i.e.,
a
natural
isomorphism
with
the
F
-vector
space
F
⊕l
given
by
the
direct
sum
of
l
copies
of
the
F
-vector
space
F
,
which
we
think
of
as
being
labeled
by
t
=
0,
.
.
.
,
l
−
1],
which
is
compatible
—
up
to
contributions
that
are
sufficiently
small
as
to
have
no
effect
on
the
computation
of
the
leading
terms
of
interest
—
with
the
various
natural
integral
structures
and
natural
Hermitian
metrics
involved,
except
at
the
valuations
where
E
has
potentially
multiplicative
reduction,
where
one
must
adjust
the
natural
integral
structure
[i.e.,
the
integral
structure
determined
by
the
ring
of
integers
O
F
of
F
]
on
the
copy
of
F
in
F
⊕l
labeled
t
∈
{0,
.
.
.
,
l
−
1}
by
a
factor
of
2
q
t
/4
—
where
the
notation
“q”
denotes
a
2l-th
root
of
the
q-parameter
of
E
at
the
valuation
under
consideration.
Next,
let
us
observe
that
[again
up
to
contributions
that
are
sufficiently
small
as
to
have
no
effect
on
the
computation
of
the
leading
terms
of
interest]
one
may
replace
the
label
t
∈
{0,
.
.
.
,
l
−
1}
by
a
label
j
∈
{1,
.
.
.
,
l
},
where
we
think
def
⊗2
of
t
as
≈
2j.
Write
Ω
log
E
=
ω
E
.
Then
the
1-dimensional
F
-vector
spaces
—
i.e.,
which
we
think
of
as
arithmetic
line
bundles
by
equipping
these
1-dimensional
F
-vector
spaces
with
natural
integral
structures
and
natural
Hermitian
metrics
—
corresponding
to
the
label
j
∈
{1,
.
.
.
,
l
}
on
the
left-
and
right-hand
sides
of
the
natural
isomorphism
of
l-
dimensional
F
-vector
spaces
determined
by
the
fundamental
theorem
of
Hodge-Arakelov
theory
assume
the
form
2
⊗j
∨
q
j
·
F
{(Ω
log
E
)
}
,
—
where
the
notation
“∨”
denotes
the
dual.
Put
another
way,
if
we
tensor
the
dual
of
the
left-hand
side
contribution
at
j
∈
{1,
.
.
.
,
l
}
with
the
right-hand
side
contribution
at
j
∈
{1,
.
.
.
,
l
},
then
we
obtain
the
conclusion
that
the
natural
isomorphism
under
consideration
may
be
thought
of
“at
a
very
rough
level”
—
i.e.,
by
replacing
the
Hodge
filtration
with
its
semi-simplification,
etc.
—
as
a
sort
of
global
section
of
some
sort
of
weighted
average
over
j
of
the
[arithmetic
line
bundles
corresponding
to
the]
1-dimensional
F
-vector
spaces
⊗j
q
j
·
(Ω
log
E
)
2
—
where
j
ranges
over
the
elements
of
{1,
.
.
.
,
l
}.
In
fact,
the
above
discussion
may
be
translated
into
purely
geometric
terms
by
working
with
the
tautological
one-dimensional
semi-abelian
scheme
over
the
natural
Shinichi
Mochizuki
120
compactification
of
the
moduli
stack
of
elliptic
curves
[over,
say,
a
field
of
characteristic
zero].
Then
“Ω
log
E
”
may
be
thought
of
as
the
line
bundle
of
logarithmic
differentials
on
this
compactified
moduli
stack.
Moreover,
one
can
compute
global
degrees
“deg(−)”
l
⊗j
deg((Ω
log
E
)
)
−
2
deg(
j
2l
·
[∞])
=
j=1
l
2
j
j
·
deg(Ω
log
E
)
−
2l
·
deg([∞])
j=1
≈
=
=
log
1
l
2
1
l
3
2
·
(
2
)
·
deg(Ω
E
)
−
3·2l
·
(
2
)
·
deg([∞])
log
l
2
l
2
8
·
deg(Ω
E
)
−
48
·
deg([∞])
log
⊗6
l
2
)
−
deg([∞])}
48
{deg((Ω
E
)
—
where
the
notation
“≈”
denotes
a
possible
omission
of
terms
that
do
not
affect
the
leading
term;
“[∞]”
denotes
the
effective
divisor
on
the
compactified
moduli
stack
under
consideration
determined
by
the
point
at
infinity,
i.e.,
the
scheme-theoretic
zero
locus
of
the
q-parameter;
by
abuse
of
notation,
we
use
the
same
notation
for
“compactified
moduli
stack
versions”
of
the
corresponding
objects
introduced
in
the
discussion
of
elliptic
curves
over
NF’s.
That
is
to
say,
in
summary,
the
determinant
of
the
natural
isomorphism
that
appears
in
the
funda-
mental
theorem
of
Hodge-Arakelov
theory
is
simply
[an
invertible
con-
stant
multiple
of]
some
positive
tensor
power
of
the
well-known
discriminant
⊗12
⊗6
=
(Ω
log
whose
modular
form
of
weight
12,
i.e.,
a
global
section
of
ω
E
E
)
unique
zero
is
a
zero
of
order
1
at
the
point
at
infinity
of
the
compactified
moduli
stack
of
elliptic
curves
[cf.
[Pano],
§1;
the
discussion
of
the
final
portion
of
[HASurI],
§1.2,
for
more
details].
(ii)
Comparison
with
inter-universal
Teichmüller
theory:
First,
we
begin
with
the
observation
that,
relative
to
the
classical
analogy
between
NF’s
and
one-
dimensional
functions
fields
[over
some
field
of
constants],
it
is
natural
to
think
of
·
log-shells
as
localized
absolute
arithmetic
analogues
of
the
notion
of
the
sheaf
of
logarithmic
differentials.
Indeed,
this
point
of
view
is
supported
by
the
fact
that
the
log-shell
associated
to
a
finite
extension
of
Q
p
[for
some
prime
number
p]
whose
absolute
ramification
index
is
≤
p
−
2
coincides
with
the
dual
fractional
ideal
to
the
different
ideal
of
the
given
finite
extension
of
Q
p
[cf.
[IUTchIV],
Proposition
1.2,
(i);
[IUTchIV],
Proposition
1.3,
(i)].
Thus,
it
is
natural
to
regard
·
the
[arithmetic
line
bundle
corresponding
to
the]
1-dimensional
F
-vector
space
⊗j
q
j
·
(Ω
log
E
)
2
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
121
—
where
j
ranges
over
the
elements
of
{1,
.
.
.
,
l
}
—
of
the
discussion
of
(i)
as
a
sort
of
scheme-theoretic
analogue,
or
precursor,
of
the
portion
labeled
by
j
of
the
multiradial
representation
discussed
in
§3.7,
(i)
[cf.
also
the
est
explicit
display
of
§3.7,
(ii),
(8
)];
·
the
resulting
computation
of
global
degrees
“deg(−)”
given
in
(i)
as
a
sort
of
scheme-theoretic
analogue,
or
precursor,
of
the
computation
of
the
leading
est
term
of
the
log-volume
of
the
left-hand
side
of
the
inequality
of
§3.7,
(ii),
(12
)
[cf.
the
final
portion
of
§3.7,
(iv)].
Indeed,
this
was
precisely
the
point
of
view
of
the
author
around
the
year
2000
that
motivated
the
author
to
develop
inter-universal
Teichmüller
theory.
(iii)
Analytic
torsion
interpretation:
In
conventional
Arakelov
theory
for
va-
rieties
over
NF’s,
analytic
torsion
refers
to
a
metric
invariant,
at
the
archimedean
valuations
of
an
NF,
that
measures
the
way
in
which
the
space
of
global
[holomor-
phic/algebraic]
sections
of
a
line
bundle
—
which
is
regarded,
by
means
of
various
considerations
in
harmonic
analysis,
as
a
subspace
of
the
Hilbert
space
of
L
2
-class
sec-
tions
of
the
line
bundle
—
is
embedded
inside
this
ambient
Hilbert
space
of
L
2
-class
sections.
Since
this
ambient
Hilbert
space
of
L
2
-class
sections
may
be
regarded
as
a
topo-
logical
invariant,
i.e.,
which
is
unaffected
by
deformations
of
the
holomorphic
moduli
of
the
variety
under
consideration,
the
notion
of
analytic
torsion
may
be
understood
as
a
measure
of
the
way
in
which
the
subspace
constituted
by
the
space
of
global
algebraic
sections
—
which
depends,
in
a
quite
essential
fashion,
on
the
holomorphic
moduli
of
the
variety
under
consideration
—
is
embedded
inside
this
topological
invariant.
When
formulated
in
this
way,
the
notion
of
analytic
torsion
becomes
highly
reminiscent
of
the
computa-
tional
technique
of
mono-analytic
transport
[cf.
the
discussion
of
§2.7;
§2.9;
§3.1,
(v)]
and,
in
particular,
of
the
use
of
log-shells
to
construct
the
“multira-
dial
containers”
[cf.
the
discussion
of
§3.6,
(iv)]
for
the
various
arithmetic
holomorphic
structures
that
appear
in
the
multiradial
representation
discussed
in
§3.7,
(i).
Indeed,
from
this
point
of
view,
scheme-theoretic
Hodge-Arakelov
theory
may
be
under-
stood
as
a
sort
of
intermediate
step
—
i.e.,
a
finite
discrete
approximation,
in
the
spirit
of
the
discussion
of
§2.14,
which
is,
moreover,
[unlike
the
classical
notion
of
ana-
lytic
torsion!]
defined
over
NF’s
—
between
the
classical
notion
of
analytic
torsion
and
inter-universal
Teichmüller
theory.
Put
another
way,
Shinichi
Mochizuki
122
·
the
natural
isomorphism
that
appears
in
the
fundamental
theorem
of
scheme-
theoretic
Hodge-Arakelov
theory
may
be
understood
as
a
sort
of
polyno-
mial-theoretic
discretization
of
the
theory
surrounding
the
classical
notion
of
analytic
torsion,
while
·
inter-universal
Teichmüller
theory
may
be
understood
as
a
sort
of
global
Galois-theoretic
version
over
NF’s
of
the
theory
surrounding
the
classical
notion
of
analytic
torsion
[cf.
the
discussion
of
[IUTchIV],
Remark
1.10.4].
§
3.10.
The
technique
of
tripodal
transport
In
the
present
§3.10,
we
re-examine
inter-universal
Teichmüller
theory
once
again,
this
time
from
the
point
of
view
of
the
technique
of
tripodal
transport.
Various
versions
of
this
technique
may
also
be
seen
in
previous
work
of
the
author
concerning
·
p-adic
Teichmüller
theory,
·
scheme-theoretic
Hodge-Arakelov
theory,
and
·
combinatorial
anabelian
geometry.
The
proof
given
by
·
Bogomolov
[cf.
[ABKP],
[Zh],
[BogIUT],
as
well
as
the
discussion
of
§4.3,
(iii),
below]
of
the
geometric
version
of
the
Szpiro
Conjecture
over
the
complex
numbers
may
also
be
re-interpreted
from
the
point
of
view
of
this
technique.
(i)
The
notion
of
tripodal
transport:
The
general
notion
of
tripodal
trans-
port
may
be
summarized
as
follows
[cf.
also
Fig.
3.21
below]:
(1
trp
(2
trp
(3
trp
)
One
starts
with
a
“nontrivial
property”
of
interest
[i.e.,
that
one
wishes
to
verify!]
associated
to
some
sort
of
given
arithmetic
holomorphic
structure
—
such
as
a
hyperbolic
curve
or
a
number
field
[cf.
the
discussion
of
§2.7,
(vii)].
)
One
observes
that
this
nontrivial
property
of
interest
[i.e.,
associated
to
the
given
arithmetic
holomorphic
structure]
may
be
derived
by
combining
a
“relatively
trivial”
property,
again
associated
to
the
given
arithmetic
holomorphic
structure,
with
some
sort
of
alternative
property
of
interest.
)
One
establishes
some
sort
of
parallel
transport
mechanism
—
which
is
typ-
ically
not
compatible
with
the
given
arithmetic
[i.e.,
scheme-/ring-theoretic!]
holo-
morphic
structure
—
that
allows
one
to
reduce
the
issue
of
verifying
the
alternative
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
123
property
of
interest
for
the
given
arithmetic
holomorphic
structure
to
a
“corre-
sponding
version”
in
the
case
of
the
tripod
[i.e.,
the
projective
line
minus
three
points]
of
this
alternative
property
of
interest.
(4
trp
(5
trp
)
One
verifies
the
alternative
property
of
interest
in
the
case
of
the
tripod.
trp
trp
trp
trp
trp
trp
)
By
combining
(1
),
(2
),
(3
),
(4
),
one
concludes
that
the
original
non-
trivial
property
of
interest
associated
to
the
given
arithmetic
holomorphic
structure
does
indeed
hold,
as
desired.
Here,
we
note
that
the
steps
(3
),
(4
)
are
often
very
closely
related,
and,
indeed,
at
times,
it
is
difficult
to
isolate
these
two
steps
from
one
another.
This
sort
of
argument
might
strike
some
readers
at
first
glance
as
“mysterious”
or
“astonishing”
in
the
sense
that
ultimately,
one
is
able
to
trp
conclude
the
original
nontrivial
property
of
interest
[cf.
(1
)]
associated
trp
to
the
given
arithmetic
holomorphic
structure
[cf.
(5
)]
despite
that
fact
that
the
nontrivial
content
of
the
argument
centers
around
the
arith-
trp
trp
metic
surrounding
the
tripod
[cf.
(3
),
(4
)],
in
sharp
contrast
to
the
fact
that
the
argument
only
requires
the
use
of
a
“relatively
trivial”
observation
trp
concerning
the
given
arithmetic
holomorphic
structure
[cf.
(2
)].
arithmetic
geometry
surrounding
the
tripod
(non-holomorphic!)
parallel
transport
←−−−−−−−−−→
given
arithmetic
holomorphic
structure
via
rigidity
properties
alternative
property
←−−−−−−−−−→
original
nontrivial
property
#
relatively
trivial
property
+
alternative
property
Fig.
3.21:
Tripodal
transport
Perhaps
it
is
most
natural
to
regard
this
sense
of
“mysteriousness”
or
“astonishment”
trp
as
a
reflection
of
the
potency
of
the
parallel
transport
mechanism
[cf.
(3
)]
that
is
employed.
This
“potency”
is,
in
many
of
the
examples
discussed
below,
derived
as
a
consequence
of
various
rigidity
properties,
such
as
anabelian
properties.
Such
rigidity
properties
may
only
be
derived
by
124
Shinichi
Mochizuki
applying
the
mechanism
of
parallel
transport
via
rigidity
properties
—
not
to
relatively
simple
“types
of
mathematical
objects”
such
as
vector
spaces
or
modules,
as
is
typically
the
case
in
classical
instances
of
parallel
transport!
—
but
rather
to
complicated
mathematical
objects
[cf.
the
discussion
of
[IUTchIV],
Remark
3.3.2],
such
as
the
sort
of
Galois
groups/étale
fundamental
groups
that
occur
in
anabelian
geometry,
i.e.,
mathematical
objects
whose
intrinsic
structure
is
sufficiently
rich
to
allow
one
to
establish
rigidity
proper-
ties
that
are
sufficiently
“potent”
to
compensate
for
the
“loss
of
structure”
that
arises
from
sacrificing
compatibility
with
classical
scheme-/ring-theoretic
structures.
Here,
we
note
that
it
is
necessary
to
sacrifice
compatibility
with
classical
scheme-/ring-
theoretic
structures
precisely
because
such
structures
typically
constitute
a
fundamen-
tal
obstruction
to
relating
the
arithmetic
surrounding
the
given
arithmetic
holomor-
phic
structure
to
the
arithmetic
surrounding
the
tripod.
A
typical
example
of
this
sort
of
“fundamental
obstruction”
may
be
seen
by
considering,
for
instance,
the
case
of
two
[scheme-theoretically!]
non-isomorphic
proper
hyperbolic
curves
over
an
algebraically
closed
field
of
characteristic
zero,
which,
nonetheless,
have
isomorphic
étale
fundamen-
tal
groups.
This
point
of
view,
i.e.,
of
overcoming
the
sort
of
“fundamental
obstruction”
to
parallel
transport
that
arises
from
imposing
the
restriction
of
working
within
a
fixed
scheme/ring
theory,
is
closely
related
to
the
introduction
of
the
notions
of
Frobenius-
like
and
étale-like
structures
—
cf.
the
discussion
of
§2.7,
(ii),
(iii);
§2.8.
(ii)
Inter-universal
Teichmüller
theory
via
tripodal
transport:
We
begin
our
discussion
by
observing
that,
when
viewed
from
the
point
of
view
of
the
notion
of
tripodal
transport,
inter-universal
Teichmüller
theory
may
be
recapitulated
as
follows:
est
trp
The
fundamental
log
volume
estimate
(12
)
[cf.
(1
)]
is
obtained
in
the
trp
est
est
argument
discussed
in
§3.7,
(ii)
[cf.
(5
)],
by
combining
[cf.
(9
),
(10
),
est
trp
(11
)]
a
relatively
simple
argument
[cf.
(2
)]
carried
out
in
the
arithmetic
est
est
holomorphic
structure
of
the
RHS
of
the
Θ-link
[cf.
(1
),
(7
)],
involving
relatively
simple
operations
such
as
the
formation
of
the
holomorphic
hull
[cf.
est
est
est
trp
(6
),
(7
),
(8
)],
with
the
parallel
transport
mechanism
[cf.
(3
),
as
well
as
the
discussion
of
§3.1,
(iv),
(v)]
furnished
by
the
multiradial
repre-
est
est
est
est
sentation
[cf.
(2
),
(3
),
(4
),
(5
)],
which
is
established
by
considering
various
properties
of
objects
[cf.
§3.4,
§3.6],
such
as
the
theta
function
on
the
trp
Tate
curve
[cf.
§3.4,
(iii),
(iv);
Fig.
3.9],
on
the
LHS
of
the
Θ-link
[cf.
(4
)].
Here,
we
recall
from
the
discussion
of
§3.4,
(iii),
(iv);
Fig.
3.9,
that
the
theory
surrounding
the
theta
function
on
the
Tate
curve
may
be
thought
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
125
of
as
a
sort
of
function-theoretic
representation
of
the
p-adic
arithmetic
geometry
of
a
copy
of
the
tripod
for
which
the
cusps
“0”
and
“∞”
are
subject
to
the
involution
symmetry
that
permutes
these
two
cusps
and
leaves
the
cusp
“1”
fixed.
Also,
we
recall
from
the
discussion
of
§3.7,
(i)
[cf.
also
the
discussion
of
the
properties
“IPL”,
“SHE”,
“APT”,
“HIS”
in
[IUTchIII],
Remark
3.11.1]
that
the
parallel
transport
mechanism
furnished
by
the
multiradial
representation
revolves
around
the
following
central
property:
the
algorithm
that
yields
the
multiradial
representation
converts
any
collection
of
input
data
[i.e.,
not
just
the
codomain
data
(a
q
),
(b
q
),
(c
q
)
of
the
Θ-link!]
that
is
isomorphic
to
the
domain
data
(a
Θ
),
(b
Θ
),
(c
Θ
)
of
the
Θ-link
—
i.e.,
in
somewhat
more
technical
terminology
[cf.
[IUTchII],
Definition
4.9,
(viii)],
any
F
×μ
-prime-strip
—
into
output
data
that
is
expressed
in
terms
of
the
arithmetic
holomorphic
structure
of
the
input
data,
i.e.,
of
the
codomain
of
the
Θ-link.
Finally,
at
a
more
technical
level,
we
recall
from
§3.3,
(vi);
§3.4,
(ii);
§3.4,
(iii),
(iv),
that
this
parallel
transport
mechanism
is
established
by
applying
·
the
theory
of
the
étale
theta
function
developed
in
[EtTh];
·
the
theory
of
[local
and
global]
mono-anabelian
reconstruction
developed
in
[AbsTopII],
[AbsTopIII].
Here,
it
is
of
interest
to
observe
that
both
the
theory
of
elliptic
cuspidalization,
which
plays
an
important
role
in
[EtTh],
and
the
theory
of
Belyi
cuspidalization,
which
plays
an
important
role
in
[AbsTopII],
[AbsTopIII],
may
be
regarded
as
essen-
tially
formal
consequences
of
the
fundamental
anabelian
results
obtained
in
[pGC].
The
rigidity
properties
developed
in
[EtTh]
also
depend,
in
a
fundamental
way,
on
the
interpretation
[i.e.,
as
rigidity
properties
of
the
desired
type!]
given
in
[EtTh]
of
the
theta
symmetries
of
the
theta
function
on
the
Tate
curve.
(iii)
p-adic
Teichmüller
theory
via
tripodal
transport:
When
viewed
from
the
point
of
view
of
the
notion
of
tripodal
transport,
a
substantial
portion
of
the
p-adic
Teichmüller
theory
of
[pOrd],
[pTch],
[pTchIn]
may
be
summarized
as
follows:
One
constructs
a
theory
of
canonical
indigenous
bundles,
canonical
Frobe-
nius
liftings,
and
associated
canonical
Galois
representations
into
P
GL
2
(−)
[for
a
suitable
“(−)”
—
cf.
[pTchIn],
Theorems
1.2,
1.4,
for
more
details]
for
trp
trp
trp
quite
general
p-adic
hyperbolic
curves
[cf.
(1
),
(2
),
(5
)]
by
establishing
trp
a
parallel
transport
mechanism
[cf.
(3
)]
that
allows
one
to
transport
126
Shinichi
Mochizuki
similar
canonical
objects
associated
to
the
tautological
family
of
elliptic
curves
trp
over
the
tripod
[cf.
(4
)].
Here,
we
recall
that,
prior
to
[pOrd],
the
existence
of
such
canonical
objects
associated
to
a
p-adic
hyperbolic
curve
was
only
known
in
the
case
of
Shimura
curves,
i.e.,
such
as
the
tripod.
From
the
point
of
view
of
the
notion
of
tripodal
transport,
it
is
also
of
interest
to
observe
that:
The
notion
of
an
ordinary
Frobenius
lifting
[cf.
[pTchIn],
Theorem
1.3],
which
plays
a
central
role
in
[pOrd],
[pTch],
[pTchIn],
may
be
understood
as
a
sort
of
p-adic
generalization
of
the
most
fundamental
example
of
a
Frobenius
lifting,
namely,
the
endomorphism
T
→
T
p
[where
T
denotes
the
standard
coordinate
on
the
projective
line]
of
the
tripod
over
a
p-adic
field.
This
endomorphism
is
equivariant
with
respect
to
the
sym-
metry
of
the
tripod
which
permutes
the
cusps
“0”
and
“∞”
and
leaves
the
cusp
“1”
fixed.
At
a
more
technical
level,
we
recall
that
the
parallel
transport
mechanism
employed
in
p-adic
Teichmüller
theory
revolves
around
the
following
two
fundamental
technical
tools:
·
the
fact
that
the
natural
morphism
from
the
moduli
stack
of
nilcurves
[i.e.,
pointed
stable
curves
equipped
with
an
indigenous
bundle
whose
p-curvature
is
square
nilpotent]
to
the
corresponding
moduli
stack
of
pointed
stable
curves
is
a
finite,
flat,
and
local
complete
intersection
morphism
of
degree
p
to
the
power
of
the
dimension
of
these
moduli
stacks
[cf.
[pTchIn],
Theorem
1.1];
·
various
strong
rigidity
properties,
with
respect
to
deformation,
that
hold
precisely
over
the
ordinary
locus
of
the
moduli
stack
of
nilcurves,
i.e.,
the
étale
locus
of
the
natural
morphism
from
the
moduli
stack
of
nilcurves
to
the
corresponding
moduli
stack
of
pointed
stable
curves.
In
this
context,
it
is
of
interest
to
observe,
considering
the
fundamental
role
played
by
such
notions
as
differentials
and
curvature
in
the
classical
differential-geometric
ver-
sion
of
parallel
transport,
that
both
of
these
fundamental
technical
tools
rely
on
various
subtle
properties
of
the
p-curvature
and
Frobenius
actions
on
differentials.
This
relationship
with
differentials
is
also
interesting
from
the
point
of
view
of
the
funda-
mental
role
played
by
the
theory
of
[pGC]
in
the
discussion
of
[EtTh],
[AbsTopII],
and
[AbsTopIII]
in
(ii),
since
differentials,
treated
from
a
p-adic
Hodge-theoretic
point
of
view,
play
a
fundamental
role
in
[pGC].
Finally,
we
observe
that
although
anabelian
results
do
not
play
any
role
in
the
parallel
transport
mechanism
of
p-adic
Teichmüller
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
127
theory,
it
is
interesting
to
note
that
p-adic
Teichmüller
theory
has
an
important
appli-
cation
to
absolute
anabelian
geometry
[cf.
[CanLift],
§3,
as
well
as
the
discussion
of
[IUTchI],
§I4;
[IUTchII],
Remark
4.11.4,
(iii)].
(iv)
Scheme-theoretic
Hodge-Arakelov
theory
via
tripodal
transport:
When
viewed
from
the
point
of
view
of
the
notion
of
tripodal
transport,
the
fundamen-
tal
theorem
of
Hodge-Arakelov
theory,
i.e.,
the
natural
isomorphism
reviewed
at
the
beginning
of
§3.9,
(i)
[cf.
also
Example
2.14.3;
[HASurI];
[HASurII]],
may
be
understood
as
follows:
One
verifies
[cf.
the
discussion
of
[HASurI],
§1.1]
that
the
natural
morphism
obtained
by
evaluating
sections
of
an
ample
line
bundle
over
the
universal
vec-
torial
extension
of
an
elliptic
curve
at
torsion
points
[cf.
the
discussion
at
the
trp
trp
beginning
of
Example
2.14.3]
is
indeed
an
isomorphism
[cf.
(1
),
(5
)]
by
verifying
that
it
is
an
isomorphism
in
the
case
of
Tate
curves
by
means
of
an
trp
explicit
computation
involving
derivatives
of
theta
functions
[cf.
(4
)]
and
then
proceeding
to
parallel
transport
this
isomorphism
in
the
case
of
Tate
curves
to
the
entire
compactified
moduli
stack
of
elliptic
curves
in
characteris-
tic
0
by
means
of
an
explicit
computation
[the
leading
term
portion
of
which
is
reviewed
in
§3.9,
(i)]
of
the
degrees
of
the
vector
bundles
on
this
compactified
moduli
stack
that
constitute
the
domain
and
codomain
of
the
natural
morphism
trp
trp
under
consideration
[cf.
(2
),
(3
)].
Here,
we
recall
from
the
discussion
of
(ii)
above;
§3.4,
(iii),
(iv);
Fig.
3.9,
that
the
theory
surrounding
the
theta
function
on
the
Tate
curve
may
be
thought
of
as
a
sort
of
function-theoretic
representation
of
the
[not
necessarily
p-
!]adic
arithmetic
geometry
of
a
copy
of
the
tripod
for
which
the
cusps
“0”
and
“∞”
are
subject
to
the
involution
symmetry
that
permutes
these
two
cusps
and
leaves
the
cusp
“1”
fixed.
In
this
context,
it
is
also
perhaps
of
interest
to
recall
that
there
is
an
alternative
approach
to
the
parallel
transport
mechanism
discussed
above
[i.e.,
computing
degrees
of
vector
bundles
on
the
compactified
moduli
stack
of
elliptic
curves],
namely,
the
parallel
trans-
port
mechanism
applied
in
the
proof
of
[HASurII],
Theorem
4.3,
which
exploits
various
special
properties
of
the
Frobenius
and
Verschiebung
morphisms
in
positive
char-
acteristic.
Finally,
we
observe
that
although
the
scheme-theoretic
Hodge-Arakelov
theory
of
[HASurI],
[HASurII]
is
not
directly
related,
in
a
logical
sense,
to
anabelian
geometry,
it
nevertheless
played
a
central
role,
as
was
discussed
in
detail
in
§3.9,
in
motivating
the
development
of
inter-universal
Teichmüller
theory,
which
may
be
understood
as
a
sort
of
reformulation
of
the
essential
content
of
the
scheme-theoretic
128
Shinichi
Mochizuki
Hodge-Arakelov
theory
of
[HASurI],
[HASurII]
via
techniques
based
on
anabelian
ge-
ometry.
(v)
Combinatorial
anabelian
geometry
via
tripodal
transport:
Let
F
be
a
number
field,
F
an
algebraic
closure
of
F
,
X
a
hyperbolic
curve
over
F
,
n
≥
1
an
integer.
Write
X
n
for
the
n-th
configuration
space
of
X
[cf.,
e.g.,
[MT],
Definition
2.1,
(i)];
Π
n
for
the
étale
fundamental
group
of
X
n
×
F
F
[for
a
suitable
choice
of
basepoint];
def
Π
=
Π
1
;
Π
tpd
for
“Π”
in
the
case
where
X
is
the
tripod
[i.e.,
the
projective
line
minus
def
three
points];
G
F
=
Gal(F
/F
);
Out
FC
(Π
n
)
for
the
group
of
outer
automorphisms
of
Π
n
satisfying
certain
technical
conditions
[i.e.,
“FC”]
involving
the
fiberwise
subgroups
and
cuspidal
inertia
subgroups
[cf.
[CombCusp],
Definition
1.1,
(ii),
for
more
details].
Thus,
it
follows
from
the
definition
of
“Out
FC
”
that
the
natural
projection
X
n+1
→
X
n
given
by
forgetting
the
(n
+
1)-st
factor
determines
a
homomorphism
φ
n+1
:
Out
FC
(Π
n+1
)
→
Out
FC
(Π
n
)
[cf.
the
situation
discussed
in
[NodNon],
Theorem
B];
the
natural
action
of
G
F
on
X
n
×
F
F
determines
an
outer
Galois
representation
ρ
n
:
G
F
→
Out
FC
(Π
n
)
def
[cf.
the
situation
discussed
in
[NodNon],
Theorem
C].
Write
ρ
=
ρ
1
,
ρ
tpd
for
“ρ”
in
the
case
where
X
is
the
tripod.
Then
the
proof
of
the
injectivity
[cf.
[NodNon],
Theorem
C]
of
ρ
:
G
F
→
Out
FC
(Π)
given
in
[NodNon]
is
perhaps
the
most
transparent/prototypical
example
of
the
phenomenon
of
tripodal
transport.
Indeed,
this
proof
may
be
summarized
as
follows:
One
makes
the
[“relatively
trivial”!
—
cf.
(2
factorization
trp
)]
observation
that
ρ
admits
a
ρ
=
φ
2
◦
φ
3
◦
ρ
3
:
G
F
→
Out
FC
(Π
3
)
→
Out
FC
(Π
2
)
→
Out
FC
(Π)
trp
—
which
allows
one
to
reduce
[cf.
(2
)]
the
verification
of
the
desired
injectivity
def
trp
trp
of
ρ
[cf.
(1
),
(5
)]
to
the
verification
of
the
injectivity
of
φ
23
=
φ
2
◦
φ
3
[cf.
trp
trp
trp
(3
),
(4
)]
and
ρ
3
[cf.
(4
)].
Then:
·
One
observes
that
the
injectivity
of
φ
23
depends
only
on
the
type
“(g,
r)”
[i.e.,
the
genus
and
number
of
punctures]
of
X,
hence
may
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
129
be
verified
in
the
case
of
—
i.e.,
may
be
“parallel
transported”
trp
[cf.
(3
)]
to
the
case
of
—
a
totally
degenerate
pointed
stable
curve,
i.e.,
a
pointed
curve
obtained
by
gluing
together
some
collec-
trp
tion
of
tripods
along
the
various
cusps
of
the
tripods
[cf.
(4
)].
On
the
other
hand,
in
the
case
of
such
a
totally
degenerate
pointed
stable
curve,
the
desired
injectivity
[i.e.,
of
the
analogue
of
“φ
23
”]
may
be
verified
by
applying
the
purely
combinatorial/group-theoretic
tech-
niques
of
combinatorial
anabelian
geometry
developed
in
[Com-
bCusp],
[NodNon]
[cf.
[NodNon],
Theorem
B].
·
One
verifies
the
injectivity
of
ρ
3
by
applying
a
certain
natural
ho-
momorphism
called
the
tripod
homomorphism
τ
:
Out
FC
(Π
3
)
→
Out
FC
(Π
tpd
)
[cf.
[CbTpII],
Theorem
C,
(ii)],
which
satisfies
the
property
that
ρ
tpd
=
τ
◦
ρ
3
:
G
F
→
Out
FC
(Π
3
)
→
Out
FC
(Π
tpd
)
and
hence
allows
one
to
conclude
the
injectivity
of
ρ
3
from
the
well-known
injectivity
trp
result
of
Belyi
to
the
effect
that
ρ
tpd
is
injective
[cf.
(4
)].
Here,
it
is
interesting
to
note,
especially
in
light
of
the
discussion
of
anabelian
results
and
differentials
in
the
final
portions
of
(i),
(ii),
(iii),
the
central
role
played
by
combinato-
rial
anabelian
geometry
—
i.e.,
in
particular,
various
combinatorial
versions
of
the
Grothendieck
Conjecture
such
as
[NodNon],
Theorem
A
—
in
the
parallel
transport
mechanism
discussed
above.
Such
combinatorial
versions
of
the
Grothendieck
Conjec-
ture
concern
group-theoretic
characterizations
of
the
decomposition
of
a
pointed
stable
curve
into
various
irreducible
components
glued
together
along
the
nodes
of
the
curve.
This
sort
of
decomposition
may
be
interpreted
as
a
sort
of
discrete
version
of
the
notion
of
a
differential,
i.e.,
which
may
be
thought
of
as
a
decomposition
of
a
ring/scheme
structure
into
infinitesimals.
Finally,
we
emphasize
that
this
proof
of
the
injectivity
of
ρ
is
a
particularly
striking
example
of
the
phenomenon
of
tripodal
transport,
in
the
sense
that
the
issue
of
relating
the
injectivity
of
ρ
for
an
arbitrary
X
to
the
injectivity
of
ρ
tpd
,
i.e.,
in
the
case
of
the
tripod,
seems,
a
priori,
to
be
entirely
intractable,
at
least
so
long
as
one
restricts
oneself
to
morphisms
between
schemes
[cf.
the
discussion
in
the
final
portion
of
(i)].
(vi)
Tripodal
transport
and
Bogomolov’s
proof:
Often,
as
in
the
examples
discussed
in
(ii),
(iii),
(iv),
above,
the
tripod
that
appears
in
instances
of
the
phe-
nomenon
of
tripodal
transport
is
a
tripod
in
which
the
cusps
“0”
and
“∞”
play
a
distinguished,
but
symmetric
role,
which
is
somewhat
different
from
the
role
played
by
the
cusp
“1”.
When
considered
from
this
point
of
view,
the
tripod
may
thought
of
as
the
underlying
scheme
of
the
group
scheme
G
m
[with
its
origin
removed],
hence,
in
Shinichi
Mochizuki
130
particular,
as
a
sort
of
algebraic
version
of
the
topological
circle
S
1
.
If
one
thinks
of
the
tripod
in
this
way,
i.e.,
as
corresponding
to
S
1
,
then
the
proof
given
by
Bogomolov
[cf.
[ABKP],
[Zh],
[BogIUT],
as
well
as
the
discussion
of
§4.3,
(iii),
below]
of
the
geometric
version
of
the
Szpiro
Conjecture
over
the
complex
num-
bers
may
also
be
understood
as
an
instance,
albeit
in
a
somewhat
generalized
sense,
of
the
technique
of
tripodal
transport.
To
explain
further,
we
introduce
notation
as
follows:
∼
·
Write
Aut
π
(R)
for
the
group
of
self-homeomorphisms
R
→
R
that
commute
with
translation
by
π
∈
R.
Thus,
if
we
think
of
S
1
as
the
quotient
R/(2π·Z),
then
Aut
π
(R)
may
be
understood
as
the
group
of
self-homeomorphisms
of
R
that
lift
elements
of
the
group
Aut
+
(S
1
)
of
orientation-preserving
self-homeomorphisms
of
S
1
that
commute
with
multiplication
by
−1
on
S
1
.
In
particular,
we
have
a
natural
exact
sequence
1
→
2π
·
Z
→
Aut
π
(R)
→
Aut
+
(S
1
)
→
1.
∼
·
Write
Aut
π
(R
≥0
)
for
the
group
of
self-homeomorphisms
R
≥0
→
R
≥0
that
stabilize
and
restrict
to
the
identity
on
the
subset
π
·
N
⊆
R
≥0
.
·
Write
R
|π|
for
the
set
of
Aut
π
(R
≥0
)-orbits
of
R
≥0
[relative
to
the
natural
action
of
Aut
π
(R
≥0
)
on
R
≥0
].
Thus,
!
!
R
|π|
=
∪
n∈N
{
[n
·
π]
}
m∈N
{
[(m
·
π,
(m
+
1)
·
π)]
}
—
where
we
use
the
notation
“[−]”
to
denote
the
element
in
R
|π|
determined
by
an
element
or
nonempty
subset
of
R
≥0
that
lies
in
a
single
Aut
π
(R
≥0
)-orbit;
we
use
the
notation
“(−,
−)”
to
denote
an
open
interval
in
R
≥0
;
we
observe
that
the
natural
order
relation
on
R
≥0
induces
a
natural
order
relation
on
R
|π|
.
·
Write
δ
sup
:
Aut
π
(R)
→
R
|π|
for
the
map
that
assigns
to
α
∈
Aut
π
(R)
the
element
sup(δ(α))
∈
R
|π|
,
where
we
observe
that
def
δ(α)
=
{
[
|α(x)
−
x|
]
|
x
∈
R
}
⊆
R
|π|
is
a
finite
subset
[cf.
the
definition
of
Aut
π
(R)!]
of
R
|π|
,
and
that
[as
is
easily
ver-
ified,
by
observing
that
for
any
β
∈
Aut
π
(R)
and
x,
y
∈
R
such
that
x
≤
y,
there
exists
a
γ
∈
Aut
π
(R
≥0
)
such
that
β(y)−β(x)
=
β((y
−x)+x)−β(x)
=
γ(y
−x)]
the
assignments
δ(−),
δ
sup
(−)
are
Aut
π
(R)-conjugacy
invariant.
·
Write
SL
2
(R)
∼
for
universal
covering
of
SL
2
(R).
Thus,
we
have
a
natural
central
extension
of
topological
groups
1
→
Z
→
SL
2
(R)
∼
→
SL
2
(R)
→
1.
By
def
composing
the
natural
embedding
S
1
→
R
2×
=
R
2
\
{(0,
0)}
with
the
natu-
def
ral
projection
R
2×
R
2∠
=
R
2×
/R
>0
,
we
obtain
a
natural
homeomorphism
∼
S
1
→
R
2∠
,
hence
[by
considering
the
natural
action
of
SL
2
(R)
on
R
2×
,
R
2∠
]
natural
actions
SL
2
(R)
S
1
;
SL
2
(R)
∼
R
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
131
[where
we
think
of
R
as
the
universal
covering
of
S
1
=
R/(2π
·
Z)],
the
latter
of
which
determines
a
natural
injective
homomorphism
SL
2
(R)
∼
→
Aut
π
(R)
[which,
at
times,
we
shall
use
to
think
of
SL
2
(R)
∼
as
a
subgroup
of
Aut
π
(R)].
We
may
assume
without
loss
of
generality
that
the
generator
“1”
of
Z
→
SL
2
(R)
∼
was
chosen
so
as
to
act
on
R
in
the
positive
direction.
def
·
Write
SL
2
(Z)
∼
=
SL
2
(R)
∼
×
SL
2
(R)
SL
2
(Z).
Thus,
we
have
a
natural
central
extension
of
discrete
groups
1
→
Z
→
SL
2
(Z)
∼
→
SL
2
(Z)
→
1.
One
shows
easily
[e.g.,
by
considering
the
discriminant
modular
form,
as
in
[BogIUT]]
that
the
abelianization
of
SL
2
(Z)
∼
is
isomorphic
to
Z,
and
hence
that
there
exists
a
unique
surjective
homomorphism
χ
:
SL
2
(Z)
∼
Z
that
maps
positive
elements
of
Z
→
SL
2
(Z)
∼
to
positive
elements
of
Z.
In
some
sense,
the
fundamental
phenomenon
that
underlies
Bogomolov’s
proof
is
the
following
elementary
fact:
Whereas
the
SL
2
(Z)-conjugacy
classes
of
the
unipotent
elements
"
#
1
m
m
def
τ
=
∈
SL
2
(Z)
0
1
differ
for
different
positive
integers
m,
the
SL
2
(R)-conjugacy
classes
of
these
elements
coincide
for
different
positive
integers
m.
In
the
context
of
Bogomolov’s
proof,
if
one
thinks
of
SL
2
(Z)
as
the
topological
fun-
damental
group
of
the
moduli
stack
of
elliptic
curves
over
the
complex
numbers,
then
such
unipotent
elements
arise
as
the
images
in
SL
2
(Z)
—
via
the
[outer]
homomorphism
induced
on
topological
fundamental
groups
by
the
classifying
morphism
associated
to
a
family
of
one-dimensional
complex
tori
over
a
hyperbolic
Riemann
surface
S
of
finite
type
—
of
the
natural
generators
of
cuspidal
inertia
groups
of
the
topological
funda-
mental
group
of
S.
In
this
situation,
the
positive
integer
m
then
corresponds
to
the
valuation
of
the
q-parameter
at
a
cusp
of
S.
Next,
we
recall
[cf.,
e.g.,
[BogIUT],
(B1)]
that
unipotent
elements
of
SL
2
(R)
admit
canonical
liftings
to
SL
2
(R)
∼
.
In
par-
ticular,
it
makes
sense
to
apply
both
δ
sup
and
χ
to
the
canonical
lifting
τ
m
∈
SL
2
(Z)
∼
of
τ
m
.
Since
χ
is
a
homomorphism,
we
have
χ(
τ
m
)
=
m
132
Shinichi
Mochizuki
[cf.,
e.g.,
[BogIUT],
(B3)].
On
the
other
hand,
since
δ
sup
(−)
is
Aut
π
(R)-
[hence,
in
particular,
SL
2
(R)
∼
-]
conjugacy
invariant,
we
have
δ
sup
(
τ
m
)
<
[π]
[cf.,
e.g.,
[BogIUT],
(B1)]
for
arbitrary
m.
It
is
precisely
by
applying
both
χ
and
δ
sup
(−)
to
a
certain
natural
relation
[arising
from
the
image
in
SL
2
(Z)
of
the
“usual
defining
relation”
of
the
topological
fundamental
group
of
S]
between
elements
∈
SL
2
(Z)
lifted
to
SL
2
(Z)
∼
that
one
is
able
to
derive
the
geometric
version
of
the
Szpiro
inequality,
that
is
to
say,
to
bound
the
height
of
the
given
family
of
one-dimensional
complex
tori
—
i.e.,
more
concretely,
in
essence,
the
sum
of
the
“m”’s
arising
from
the
various
cusps
of
S
[cf.,
e.g.,
[BogIUT],
(B4)]
—
by
a
number
that
depends
only
on
the
genus
and
number
of
cusps
of
S
and
not
on
the
“m”’s
themselves
[cf.,
e.g.,
[BogIUT],
(B2),
(B5)].
From
the
point
of
view
of
the
technique
of
tripodal
transport,
one
may
summarize
this
argument
as
follows:
one
bounds
the
height
[i.e.,
essentially,
the
sum
of
the
“m”’s]
of
the
given
trp
family
of
one-dimensional
complex
tori
[cf.
(5
)]
—
which
is
a
reflection
trp
of
the
holomorphic
moduli
of
this
family
[cf.
(1
)]
—
by
combining
a
trp
“relatively
trivial”
[cf.
(2
)]
object
χ
arising
from
the
holomorphic
struc-
ture
of
the
moduli
stack
of
elliptic
curves
over
the
complex
numbers
[i.e.,
from
the
discriminant
modular
form]
with
the
parallel
transport
mecha-
trp
nism
[cf.
(3
)]
given
by
passing
from
the
“holomorphic”
SL
2
(Z),
SL
2
(Z)
∼
to
the
“real
analytic”
SL
2
(R),
SL
2
(R)
∼
,
i.e.,
in
essence,
by
passing
to
the
Aut
+
(S
1
)-invariant
geometry
of
S
1
,
as
reflected
in
the
Aut
π
(R)-conjugacy
trp
invariant
map
δ
sup
:
Aut
π
(R)
→
R
|π|
[cf.
(4
)].
From
the
point
of
view
of
the
analogy
[cf.
the
discussion
of
(ii)
above;
[BogIUT]]
between
Bogomolov’s
proof
and
inter-universal
Teichmüller
theory,
we
observe
that:
·
The
canonical
lifts
discussed
above
of
unipotent
elements
∈
SL
2
(Z)
to
SL
2
(Z)
∼
correspond
to
the
theory
of
the
étale
theta
function
[i.e.,
[EtTh]]
in
inter-universal
Teichmüller
theory.
·
The
Aut
+
(S
1
)-invariant
geometry
of
S
1
,
as
reflected
in
the
Aut
π
(R)-
conjugacy
invariant
map
δ
sup
:
Aut
π
(R)
→
R
|π|
,
corresponds
to
the
theory
est
of
mono-analytic
log-shells
and
related
log-volume
estimates
[cf.
(12
);
§3.7,
(iv);
[IUTchIV],
§1,
§2]
in
inter-universal
Teichmüller
theory.
In
partic-
ular,
Aut
+
(S
1
)-/Aut
π
(R)-indeterminacies
in
Bogomolov’s
proof
—
in
which
both
the
additive
[i.e.,
corresponding
to
unipotent
subgroups
of
SL
2
(R)]
and
multiplicative
[i.e.,
corresponding
to
toral,
or
equivalently,
compact
subgroups
of
SL
2
(R)]
dimensions
of
SL
2
(R)
are
“confused”
within
the
single
dimen-
sion
of
S
1
—
correspond
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
of
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
133
inter-universal
Teichmüller
theory.
·
The
role
played
by
SL
2
(Z),
SL
2
(Z)
∼
,
χ
corresponds
to
the
role
played
by
the
fixed
arithmetic
holomorphic
structure
of
the
RHS
of
the
Θ-link
[cf.
est
est
est
est
est
est
est
(1
),
(6
),
(7
),
(8
),
(9
),
(10
),
(11
)]
in
the
argument
of
§3.7,
(ii).
By
contrast,
the
role
played
by
SL
2
(R),
SL
2
(R)
∼
,
δ
sup
(−)
corresponds
to
the
est
est
est
est
role
played
by
the
multiradial
representation
[cf.
(2
),
(3
),
(4
),
(5
)]
in
the
argument
of
§3.7,
(ii).
In
particular,
one
has
natural
correspondences
SL
2
(R),
SL
2
(R)
∼
,
δ
sup
(−)
SL
2
(Z),
SL
2
(Z)
∼
,
χ
←→
[IUTchIII],
Theorem
3.11;
←→
[IUTchIII],
Corollary
3.12
(⇐=
Theorem
3.11)
—
i.e.,
where,
more
precisely,
the
RHS
of
the
latter
correspondence
is
to
be
under-
stood
as
referring
to
the
derivation
of
[IUTchIII],
Corollary
3.12,
from
[IUTchIII],
The-
orem
3.11.
These
last
two
correspondences
are
particularly
interesting
in
light
of
the
well-documented
historical
fact
that
the
theory/estimates
in
Bogomolov’s
proof
related
to
SL
2
(R),
SL
2
(R)
∼
,
δ
sup
(−)
were
apparently
already
known
to
Milnor
in
the
1950’s
[cf.
[MlWd]],
while
the
idea
of
combining
these
estimates
with
the
theory
surround-
ing
SL
2
(Z),
SL
2
(Z)
∼
,
χ
appears
to
have
been
unknown
until
the
work
of
Bogomolov
around
the
year
2000
[cf.
[ABKP]].
Moreover,
these
last
two
correspondences
—
and,
indeed,
the
entire
analogy
between
Bogomolov’s
proof
and
inter-universal
Teichmüller
theory
—
are
also
of
interest
in
the
following
sense:
Bogomolov’s
proof
only
involves
working
with
elements
∈
SL
2
(R),
SL
2
(R)
∼
that
arise
from
topological
fundamental
groups,
hence
may
be
applied
not
only
to
algebraic/holomorphic
families
of
elliptic
curves,
but
also
to
arbitrary
topolog-
ical
families
of
one-dimensional
complex
tori
that
satisfy
suitable
conditions
at
the
points
of
degeneration,
i.e.,
“bad
reduction”.
This
aspect
of
Bogomolov’s
proof
is
reminiscent
of
the
fact
that
the
initial
Θ-data
of
inter-universal
Teichmüller
theory
[cf.
§3.3,
(i)]
essentially
only
involves
data
that
arises
from
various
arithmetic
fundamental
groups
associated
to
an
elliptic
curve
over
a
number
field.
In
particular,
this
aspect
of
Bogomolov’s
proof
suggests
strongly
that
perhaps,
in
the
future,
some
version
of
inter-universal
Teichmüller
theory
could
be
de-
veloped
in
which
the
initial
Θ-data
of
the
current
version
of
inter-universal
Teichmüller
theory
is
replaced
by
some
collection
of
topological
groups
that
satisfies
conditions
anal-
ogous
to
the
conditions
satisfied
by
the
collection
of
arithmetic
fundamental
groups
that
appear
in
the
initial
Θ-data
of
the
current
version
of
inter-universal
Teichmüller
theory,
but
that
does
not
necessarily
arise,
in
a
literal
sense,
from
an
elliptic
curve
over
a
number
field.
Shinichi
Mochizuki
134
§
3.11.
Mathematical
analysis
of
elementary
conceptual
discomfort
We
conclude
our
exposition,
in
the
present
§3,
of
the
main
ideas
of
inter-universal
Teichmüller
theory
by
returning
to
our
discussion
of
the
point
of
view
of
a
hypothetical
high-school
student,
in
the
style
of
§1.
Often
the
sort
of
deep
conceptual
discom-
fort
that
such
a
hypothetical
high-school
student
might
experience
when
attempting
to
understand
various
elementary
ideas
in
mathematics
may
be
analyzed
and
eluci-
dated
more
constructively
when
viewed
from
the
more
sophisticated
point
of
view
of
a
professional
mathematician.
Moreover,
this
sort
of
approach
to
mathematical
anal-
ysis
of
conceptual
discomfort
may
be
applied
to
the
analysis
of
the
discomfort
that
some
mathematicians
appear
to
have
experienced
when
studying
various
central
ideas
of
inter-universal
Teichmüller
theory,
such
as
the
log-
and
Θ-links.
(i)
Proof
by
mathematical
induction:
Some
high-school
students
encounter
substantial
discomfort
in
accepting
the
notion
of
proof
by
mathematical
induction,
for
instance,
in
the
case
of
proofs
of
facts
such
as
the
following:
Example
3.11.1.
integer
n,
it
holds
that
Sum
of
squares
of
consecutive
integers.
For
any
positive
n
j
2
=
1
6
n(2n
+
1)(n
+
1).
j=1
Such
discomfort
is
at
times
expressed
by
assertions
to
the
effect
that
they
cannot
believe
that
it
is
not
possible
to
simply
give
some
sort
of
more
direct
argument
that
applies
to
all
positive
integers
at
once
—
i.e.,
without
resorting
to
such
indirect
and
“non-
intuitive”
devices
of
reasoning
as
the
induction
hypothesis
—
in
the
style
of
proofs
of
facts
such
as
the
following:
Example
3.11.2.
holds
that
Square
of
a
sum.
For
any
positive
integers
n
and
m,
it
(n
+
m)
2
=
n
2
+
2nm
+
m
2
.
On
the
other
hand,
from
the
more
sophisticated
point
of
view
of
a
professional
mathe-
matician,
the
situation
surrounding
the
usual
proofs
of
these
facts
in
Examples
3.11.1,
3.11.2
may
be
understood
as
follows:
The
fact
in
Example
3.11.2
in
fact
holds
for
arbitrary
elements
“n”
and
“m”
in
an
arbitrary
commutative
ring
and
hence,
in
particular,
is
best
understood
as
a
consequence
of
the
axioms
of
a
commutative
ring.
By
contrast,
the
fact
in
Example
3.11.1
is
a
fact
that
depends
on
the
structure
of
the
particular
ring
Z,
or,
essentially
equivalently,
on
the
structure
of
the
particular
monoid
N.
In
particular,
it
is
natural
that
any
proof
of
the
fact
in
Example
3.11.1
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
135
should
depend,
in
an
essential
way,
on
the
definition
of
[Z
or]
N.
On
the
other
hand,
the
logical
structure
of
an
argument
by
mathematical
induction
is,
in
essence,
simply
a
rephrasing
of
the
very
definition
of
N.
In
light
of
this
state
of
affairs,
although
it
seems
to
be
rather
difficult
to
formulate
and
prove,
in
a
rigorous
way,
the
assertion
that
there
does
not
exist
a
proof
of
the
fact
in
Example
3.11.1
that
does
not
essentially
involve
mathematical
induction,
at
least
from
the
standard
point
of
view
of
mathematics
at,
say,
the
undergraduate
or
graduate
level,
it
does
not
seem
natural
or
reasonable
to
expect
the
existence
of
a
proof
of
the
fact
in
Example
3.11.1
that
does
not
essentially
involve
mathematical
induction.
(ii)
Identification
of
the
domain
and
codomain
of
the
logarithm:
A
some-
what
different
situation
from
the
situation
discussed
in
(i)
may
be
seen
in
the
case
of
the
notion
of
a
logarithm.
Some
high-school
students
encounter
substantial
discomfort
in
accepting
the
notion
of
a
logarithm
on
the
grounds
that
a
number
in
the
exponent
of
an
expression
such
as
a
b
[where,
say,
a,
b
∈
R
>0
],
i.e.,
“b”,
seems
to
have
a
“fundamentally
different
meaning”
from
a
number
not
in
the
exponent,
i.e.,
“a”.
In
light
of
this
“fundamental
difference
in
meaning”
between
numbers
in
and
not
in
the
exponent,
a
function
such
as
the
logarithm,
i.e.,
which
“converts”
[cf.
such
relations
as
log(a
b
)
=
b
·
log(a)]
numbers
in
the
exponent
into
numbers
not
in
the
exponent,
seems,
from
the
point
of
view
of
such
students,
to
be
“infinitely
mysterious”
or
intractable
in
nature.
From
the
point
of
view
of
a
professional
mathematician,
this
sort
of
“fundamental
difference
in
meaning”
between
numbers
in
and
not
in
the
exponent
may
be
understood
as
the
difference
between
iteration
of
the
monoid
operation
in
the
underlying
multiplicative
and
additive
monoids
of
the
topological
field
R,
i.e.,
as
the
difference
between
the
multiplicative
and
additive
structures
of
the
topological
field
R.
The
[natural]
logarithm
on
positive
real
numbers
may
then
be
understood
as
a
certain
natural
isomorphism
∼
log
:
R
>0
→
R
between
the
underlying
[positive]
multiplicative
and
additive
monoids
of
the
topological
field
R.
Thus,
the
substantial
discomfort
that
some
high-school
students
encounter
in
accepting
the
notion
of
a
logarithm
may
be
understood
as
a
difficulty
in
accepting
the
identification
of
the
underlying
additive
monoid
of
the
topological
field
R
that
contains
the
multiplicative
monoid
R
>0
in
the
domain
of
log
with
the
underlying
additive
monoid
of
the
topological
field
R
that
appears
in
the
codomain
of
log
on
the
grounds
that
the
map
log
is
not
136
Shinichi
Mochizuki
compatible
with
[i.e.,
does
not
arise
from
a
ring
homomorphism
between]
the
ring
structures
in
its
domain
and
codomain.
Here,
we
recall
that
this
identification
is
typically
“taken
for
granted”
or
“regarded
as
not
requiring
any
justification”
in
discussions
concerning
the
natural
logarithm
on
positive
real
numbers.
A
similar
identification
that
is
“taken
for
granted”
or
“regarded
as
not
requiring
any
justification”
may
be
seen
in
typical
discussions
concerning
the
p-adic
logarithm,
as
well
as
in
the
closely
related
identification,
in
the
context
of
p-adic
Hodge
theory,
of
the
copy
of
“Z
p
”
lying
inside
the
base
field
of
a
p-adic
variety
with
the
copy
of
“Z
p
”
that
acts
on
certain
types
of
étale
local
systems
on
the
variety
[cf.
the
discussion
of
[EtTh],
Remark
2.16.2;
the
discussion
in
the
final
portion
of
[Pano],
§3;
the
discussion
of
“mysterious
tensor
products”
in
[BogIUT]].
By
contrast,
in
the
case
of
the
log-link
in
inter-universal
Teichmüller
theory,
it
is
of
crucial
importance,
as
discussed
in
the
latter
portion
of
§3.3,
(ii),
to
distinguish
the
domain
and
codomain
of
the
log-link,
since
confusion
of
the
domain
and
codomain
of
the
log-link
—
i.e.,
confusion
of
the
multiplicative
and
additive
structures
that
occur
in
the
domain
of
the
Θ-link
—
would
yield
a
situation
in
which
the
Θ-link
is
not
well-defined.
In
particular,
interestingly
enough,
although
the
substantial
discomfort
that
some
high-school
students
experience
when
studying
the
logarithm
is
inconsistent
with
the
point
of
view
typically
taken
in
discussions
of
the
natural
logarithm
on
positive
real
numbers
or
the
p-adic
logarithm
in
the
context
of
p-adic
Hodge
theory,
this
substantial
discomfort
of
some
high-school
students
is,
somewhat
remarkably,
consistent
with
the
situation
surrounding
the
log-link
in
the
context
of
the
log-theta-lattice
in
inter-
universal
Teichmüller
theory.
(iii)
Conceptual
content
of
the
ABC
inequality:
Yet
another
kind
of
situa-
tion
—
which
resembles,
in
certain
aspects,
the
situation
discussed
in
(i),
but
is
related,
in
other
aspects,
to
the
situation
discussed
in
(ii)
—
may
be
seen
in
elementary
discus-
sions
of
the
ABC
inequality
for
rational
integers
[i.e.,
an
immediate
consequence
of
the
Szpiro
Conjecture
inequality
discussed
in
§3.7,
(iv)].
This
most
fundamental
version
of
the
ABC
inequality
may
be
stated
as
follows:
There
exists
a
positive
real
number
λ
such
that
for
all
triples
(a,
b,
c)
of
relatively
prime
positive
integers
satisfying
a
+
b
=
c,
it
holds
that
λ
p
.
abc
≤
p|abc
Here,
we
observe
that
whereas
the
left-hand
side
“LHS”
of
this
inequality
is
a
quantity
that
measures
the
size
—
i.e.,
from
a
more
advanced
point
of
view,
the
height
—
of
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
137
the
triple
(a,
b,
c)
relative
to
the
additive
structure
of
the
additive
monoid
N,
the
right-hand
side
“RHS”
of
this
inequality
is
a
quantity
that
arises
from
thinking
of
the
triple
(a,
b,
c)
in
terms
of
the
multiplicative
monoid
N
≥1
modulo
the
quotient
relation
“p
∼
p
n
”
[for
n
a
positive
integer]
that
identifies
primes
with
arbitrary
positive
powers
of
primes.
Note
that
since
the
multiplicative
structure
of
N
≥1
may
be
derived
immediately
from
the
additive
structure
of
N
—
e.g.,
by
thinking
of
“a
·
b”
as
the
sum
a
+
·
·
·
+
a
of
b
copies
of
a
—
one
may
also
think
of
the
LHS
of
the
above
inequality
as
a
measure
of
the
size
of
the
triple
(a,
b,
c)
relative
to
the
ring
structure
of
Z
[i.e.,
which
involves
both
the
additive
and
multiplicative
structures
of
Z].
In
particular,
one
may
understand,
from
a
more
conceptual
point
of
view,
the
“trivial
inequality”
in
the
opposite
direction,
i.e.,
p
abc
≥
p|abc
as
a
reflection
of
the
elementary
observation
that
the
multiplicative
monoid
N
≥1
con-
sidered
modulo
“∼”
may
be
“easily
derived
from”
—
or,
in
other
words,
is
“domi-
nated/controlled
by”
—
the
additive
structure/ring
structure
of
Z:
additive
structure/ring
structure
of
Z
multiplicative
monoid
N
≥1
modulo
∼
.
By
contrast,
the
fundamental
form
of
the
ABC
inequality
recalled
above
may
be
under-
stood,
at
a
more
conceptual
level,
as
the
assertion
that,
up
to
a
certain
“indeterminacy”
[corresponding
to
“λ”],
the
multiplicative
monoid
N
≥1
considered
modulo
“∼”
is
suffi-
ciently
potent
as
to
dominate/control
the
additive
structure/ring
structure
of
Z:
additive
structure/ring
structure
of
Z
≺
multiplicative
monoid
N
≥1
modulo
∼
—
a
somewhat
startling
assertion,
since,
at
least
from
an
a
priori
point
of
view,
passing
from,
say,
the
ring
Z
to
the
multiplicative
monoid
N
≥1
modulo
∼
appears
to
involve
quite
a
substantial
loss
of
data/structure.
On
the
other
hand,
this
conceptual
interpretation
of
the
ABC
inequality
is
remarkably
reminiscent
of
the
Θ-link
[cf.
the
discussion
of
the
log-link
in
(ii)]
and
multiradial
representation
of
inter-universal
Teichmüller
theory,
which
in
effect
assert
[cf.
the
discussion
of
§3.7,
(i)]
that
the
multiplicative
monoids/Frobenioids,
together
with
Galois
actions,
that
appear
in
the
data
glued
together
via
the
Θ-link
are
sufficiently
potent
as
to
dominate/control
up
to
certain
indeterminacies
—
via
the
multiradial
rep-
resentation
of
the
Θ-pilot
—
the
ring
structure/
arithmetic
holomorphic
structure
in
the
domain
of
the
Θ-link,
i.e.,
138
Shinichi
Mochizuki
ring
structure
that
gives
to
rise
to
Θ-pilot
≺
multiplicative
monoids/Frobenioids
and
Galois
actions
in
Θ-link
.
That
is
to
say,
in
summary,
the
conceptual
interpretation
of
the
ABC
inequality
discussed
above
is
already
sufficiently
rich
as
to
strongly
suggest
numerous
characteristic
features
of
inter-universal
Teichmüller
theory,
i.e.,
such
as
the
multiradial
represen-
tation,
up
to
suitable
indeterminacies,
of
various
distinct
ring
structures
related
by
the
much
weaker
[a
priori]
data
—
i.e.,
multiplicative
monoids/
Frobenioids
and
Galois
actions
—
that
appears
in
the
Θ-link.
In
particular,
from
the
point
of
view
of
this
conceptual
interpretation
of
the
ABC
inequality,
such
characteristic
features
of
inter-universal
Teichmüller
theory
are
quite
natural
and
indeed
appear
remarkably
close
to
being
“inevitable”
in
some
suitable
sense
[cf.
the
discussion
of
(i)].
Finally,
we
observe
that
this
interplay,
with
regard
to
“dom-
ination/control”,
between
rigid
ring
structures
[which
determine
the
“height”]
and
weaker
structures
with
indeterminacies
[for
which
primes
are
identified
with
their
positive
powers]
via
the
multiradial
representation
is
remarkably
reminiscent
of
the
discussion
in
§3.10,
(vi),
of
the
interplay,
in
the
context
of
Bogomolov’s
proof
of
the
geometric
version
of
the
Szpiro
Conjecture,
between
the
rigid
SL
2
(Z)
[where
conju-
gacy
classes
of
unipotent
elements
determine
the
“height”]
and
the
less
rigid
SL
2
(R)
[where
conjugacy
classes
of
unipotent
elements
identify
arbitrary
positive
powers
of
such
elements]
via
“δ
sup
(−)”.
(iv)
Logical
AND
vs.
logical
OR,
multiple
copies,
and
multiradiality:
As
discussed
at
the
beginning
of
§3.3,
(ii),
each
lattice
point
in
the
log-theta-lattice
[i.e.,
each
“•”
in
Fig.
3.6]
represents
a
Θ
±ell
N
F
-Hodge
theater,
which
may
be
thought
of
as
a
sort
of
miniature
model
of
the
conventional
scheme/ring
theory
surrounding
the
given
initial
Θ-data.
In
the
following
discussion,
we
shall
use
the
notation
“
(−)
R”,
where
(−)
∈
{†,
‡},
to
denote
a
particular
such
model
of
conventional
scheme/ring
theory;
we
shall
write
“∗”
for
some
F
×μ
-prime-strip
[i.e.,
some
collection
of
data
that
is
isomor-
phic
either
to
the
domain
data
(a
Θ
),
(b
Θ
),
(c
Θ
)
or,
equivalently,
to
the
codomain
data
(a
q
),
(b
q
),
(c
q
)
of
the
Θ-link
—
cf.
the
discussion
of
the
latter
portion
of
§3.7,
(i);
the
discussion
of
§3.10,
(ii);
[IUTchII],
Definition
4.9,
(viii)],
regarded
up
to
isomorphism.
Thus,
the
Θ-link
may
be
thought
of
as
consisting
of
the
assignments
∗
→
‡
q
N
∈
‡
R;
∗
→
†
q
∈
†
R
—
where
N
=
1,
2,
.
.
.
,
j
2
,
.
.
.
,
(l
)
2
;
“
‡
q
N
”
denotes
the
domain
data
(a
Θ
),
(b
Θ
),
(c
Θ
)
of
the
Θ-link,
which
belongs
[i.e.,
“∈”]
to
the
model
of
conventional
scheme/ring
theory
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
139
“
‡
R”
in
the
domain
of
the
Θ-link;
“
†
q”
denotes
the
codomain
data
(a
q
),
(b
q
),
(c
q
)
of
the
Θ-link,
which
belongs
[i.e.,
“∈”]
to
the
model
of
conventional
scheme/ring
theory
“
†
R”
in
the
codomain
of
the
Θ-link.
In
this
context,
we
observe
that
one
fundamental
—
but
entirely
elementary!
—
issue
that
arises
when
considering
these
two
assignments
“∗
→
‡
q
N
”,
“∗
→
†
q”
is
the
issue
of
what
precisely
is
the
logical
relationship
between
these
two
assignments
that
constitute
the
Θ-link?
In
a
word,
the
answer
to
this
question
—
which
underlies,
in
an
essential
way,
the
entire
logical
structure
of
inter-universal
Teichmüller
theory
—
is
as
follows:
the
Θ-link
is
to
be
understood
—
as
a
matter
of
definition!
—
as
a
construction
with
respect
to
which
these
two
assignments
are
simultaneously
valid,
that
is
to
say,
from
the
point
of
view
of
symbolic
logical
relators,
∗
→
‡
q
N
∧
∗
→
†
q
—
i.e.,
where
“∧”
denotes
the
logical
relator
“AND”
[cf.
the
discussion
of
the
“distinct
labels
approach”,
“∧”
in
[IUTchIII],
Remark
3.11.1,
(vii);
[IUTchIII],
Remark
3.12.2,
(ii),
(c
itw
),
(f
itw
)].
Here,
we
observe
that:
(1
and
(2
and
)
If
one
forgets
the
distinct
labels
“‡”,
“†”,
then
the
resulting
collections
of
data
q
N
∈
R,
q
∈
R
are
different.
In
particular,
if
one
deletes
the
distinct
labels
“‡”,
“†”,
then
the
crucial
logical
relator
“∧”
no
longer
holds
[i.e.,
leads
immediately
to
a
contradiction!].
That
is
to
say,
it
is
precisely
by
distinguishing
the
two
copies
“
‡
R”,
“
†
R”
that
one
obtains
a
well-defined
construction
of
the
Θ-link,
as
described
above.
This
situation
is
reminiscent
of
the
discussion
of
distinct
copies
in
§1.3.
)
One
way
to
understand
the
notion
of
multiradiality
in
the
case
of
the
multiradial
representation
of
the
Θ-pilot
is
as
the
[highly
nontrivial!]
property
of
an
algo-
rithm
that
allows
one
to
maintain
the
validity
of
this
crucial
logical
relator
“∧”
throughout
the
execution
of
the
algorithm
—
cf.
the
discussion
of
“simultaneous
execution/meaningfulness”
in
§2.9;
§3.4,
(i);
§3.7,
(i),
as
well
as
the
discussion
of
the
properties
“IPL”,
“SHE”,
“APT”,
“HIS”
in
[IUTchIII],
Remark
3.11.1.
This
∼
point
of
view
is
reminiscent
of
the
single
vector
bundle
“p
∗
1
F|
S
log
→
p
∗
2
F|
S
log
”
KS
δ
δ
of
§3.1,
(v),
(3
)
[i.e.,
which
serves
simultaneously
as
a
pull-back
via
p
1
and
as
a
pull-back
via
p
2
!],
as
well
as
of
the
discussion
of
§1.4,
§1.5,
§1.6
[i.e.,
of
integration
on
R
2
,
as
opposed
to
R].
Shinichi
Mochizuki
140
(3
and
and
)
It
is
precisely
by
applying
this
interpretation
[cf.
(2
)]
of
multiradiality
—
i.e.,
maintenance
of
the
validity
of
this
crucial
logical
relator
“∧”
throughout
the
ex-
est
ecution
of
the
multiradial
algorithm
—
that
one
may
conclude
[cf.
§3.7,
(ii),
(10
),
est
est
(11
),
(12
)],
in
an
essentially
formal
fashion,
that
the
multiradial
representation
of
the
Θ-pilot,
regarded
up
to
suitable
indeterminacies,
is
—
simultaneously
[cf.
“∧”!]
—
a
representation
of
the
original
q-pilot
in
the
codomain
of
the
Θ-link.
Put
another
way,
one
fundamental
cause
of
certain
frequently
[and,
at
times,
some-
what
vociferously!]
articulated
misunderstandings
of
inter-universal
Teichmüller
theory
is
precisely
the
misunderstanding
that
the
Θ-link
is
to
be
understood
—
as
a
matter
of
definition!
—
as
a
construction
with
respect
to
which
the
two
assignments
“∗
→
‡
q
N
”,
“∗
→
†
q”
are
not
necessarily
required
to
be
simultaneously
valid,
that
is
to
say,
from
the
point
of
view
of
symbolic
logical
relators,
∗
→
‡
q
N
∨
∗
→
†
q
—
i.e.,
where
“∨”
denotes
the
logical
relator
“OR”.
Here,
we
observe
that,
if
one
takes
the
point
of
view
of
this
misunderstanding,
then:
or
(1
)
The
logical
relator
“∨”
remains
valid
even
if
one
forgets
the
distinct
labels
“‡”,
“†”.
In
particular,
the
use
of
distinct
copies
throughout
inter-universal
Te-
ichmüller
theory
seems
entirely
superfluous
—
cf.
the
discussion
of
distinct
copies
in
§1.3.
or
(2
)
The
multiradial
representation
of
the
Θ-pilot
—
whose
nontriviality
lies
pre-
cisely
in
the
maintenance
of
the
validity
of
the
crucial
logical
relator
“∧”
throughout
and
and
the
execution
of
the
multiradial
algorithm!
[cf.
(2
),
(3
)]
—
appears
to
be
valid
[which
is
not
surprising
since,
in
general,
∧
=⇒
∨
!],
but
entirely
devoid
of
any
interesting
content.
or
or
est
(3
)
As
a
result
of
the
point
of
view
of
(2
),
the
conclusion
[cf.
§3.7,
(ii),
(10
),
est
est
(11
),
(12
)],
in
an
essentially
formal
fashion,
that
the
multiradial
representation
of
the
Θ-pilot,
regarded
up
to
suitable
indeterminacies,
is
—
simultaneously
[i.e.,
“∧”!]
—
a
representation
of
the
original
q-pilot
in
the
codomain
of
the
Θ-link
appears
somewhat
abrupt,
mysterious,
or
entirely
unjustified.
Thus,
in
summary,
confusion,
in
the
context
of
the
Θ-link
and
the
multiradial
rep-
resentation
of
the
Θ-pilot,
between
the
logical
relators
“∧”
and
“∨”
—
which
is,
in
essence,
an
entirely
elementary
issue
[cf.
the
discussion
of
§1.3,
as
well
as
of
[IUTchIII],
Remark
3.11.1,
(vii);
[IUTchIII],
Remark
3.12.2,
(ii),
(c
toy
),
(f
toy
)]
—
has
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
141
the
potential
to
lead
to
very
deep
repercussions
with
regard
to
understanding
the
essential
logical
structure
of
inter-universal
Teichmüller
theory.
Finally,
we
conclude
the
present
discussion
by
observing
that
one
way
to
approach
the
task
of
understanding
these
aspects
of
the
essential
logical
structure
of
inter-universal
Teichmüller
theory
is
by
considering
the
following
elementary
combinatorial
and
numerical
models:
Example
3.11.3.
Elementary
combinatorial
model
of
“∧
vs.
“∨”,
mul-
tiple
copies,
and
multiradiality.
The
ideas
discussed
in
the
present
§3.11,
(iv),
may
be
summarized/expressed
simply,
in
terms
of
elementary
combinatorics,
as
in
Fig.
3.22
below.
Here,
“∗”
corresponds,
in
the
above
discussion,
to
“∗”,
i.e.,
to
some
abstract
F
×μ
-prime-strip;
each
of
the
boxes
in
the
upper
left-
and
right-hand
corners
of
the
diagram
corresponds,
in
the
above
discussion,
to
“
(−)
R”,
where
(−)
∈
{†,
‡},
i.e.,
to
some
abstract
Θ
±ell
N
F
-Hodge
theater;
each
of
these
boxes
is
equipped
with
two
distinct
substructures
“
(−)
0”,
“
(−)
1”
[which
may
be
thought
of
as
corresponding,
in
the
above
discussion,
respectively,
to
(−)
q,
(−)
q
N
]
such
that
∗
is
glued
[cf.
the
horizontal
arrows
emanating
from
either
side
of
∗]
to
‡
1,
†
0;
the
lower
box
in
the
center
is
to
be
understood
as
a
copy
of
either
of
the
two
upper
boxes
on
the
left
and
right
whose
relationship
to
∗
is,
by
definition,
indeterminate,
i.e.,
∗
corresponds
either
to
◦
0
or
to
◦
1;
the
diagonal
arrows
on
the
left
and
right
then
correspond
to
the
operation
of
forgetting
the
datum
of
which
of
“
(−)
0”,
“
(−)
1”
is
glued
to
∗.
Thus,
the
central
portion
of
Fig.
3.22,
delimited
by
dotted
lines
on
either
side,
is
to
be
thought
of
as
containing
objects
that
are,
by
definition,
neutral/symmetric
with
respect
to
the
portion
marked
with
a
“‡”
on
the
left
and
the
portion
marked
with
a
“†”
on
the
right.
Here,
the
gluings
of
∗
in
the
upper
portion
of
the
diagram
are
to
be
understood
as
being
[by
definition!]
simultaneously
valid,
i.e.,
∧
∗
→
†
0
∗
→
‡
1
—
a
situation
that
is
consistent/well-defined
precisely
because
the
two
labels
“†”
and
“‡”
and
are
regarded
as
being
distinct
[cf.
(1
)].
In
particular,
this
upper
“∧”
portion
of
the
diagram
may
be
regarded
as
a
sort
of
tautological,
or
initial,
multiradial
algorithm
[cf.
and
(2
)],
which
is
essentially
equivalent
to
the
“distinct
labels
approach”
discussed
in
[IUTchIII],
Remark
3.11.1,
(vii).
Then
the
operation
of
passing,
via
the
diagonal
arrows,
from
the
upper
“∧”
portion
of
the
diagram
to
the
lower
central
box
—
i.e.,
to
∨
∗
→
◦
0
∗
→
◦
1
—
may
be
understood
as
corresponding
to
the
“forced
identification
approach”
discussed
in
[IUTchIII],
Remark
3.11.1,
(vii),
or
[alternatively
and
essentially
equiva-
lently!],
from
the
point
of
view
of
the
above
discussion,
as
corresponding
to
the
passage
from
“∧”
to
“∨”
[where
we
recall
that,
in
general,
∧
=⇒
∨],
i.e.,
to
∨
∗
→
†
0
∗
→
‡
1
Shinichi
Mochizuki
142
or
or
[cf.
(1
),
(2
)].
Here,
we
observe
that
this
“∨”
approach
may
also
be
regarded
as
a
sort
and
or
of
“trivial
multiradial
algorithm”
[cf.
(2
),
(2
)],
i.e.,
in
the
sense,
that,
in
general,
it
holds
that
A
∨
B
is
equivalent
to
(A
∨
B)
∧
(A
∨
B).
‡
0
‡
1
..
.
‡
1
←−−−
..
.
..
.
.
..
..
.
..
.
..
.
−
−
−
→
†
0
∈
..
.
..
.
..
.
..
.
..
.
“∧”
∗
◦
◦
0
1
“∨”
†
†
0
1
Fig.
3.22:
Elementary
combinatorial
model
of
the
Θ-link
Example
3.11.4.
Elementary
numerical
model
of
“∧
vs.
“∨”,
multiple
copies,
and
multiradiality.
Alternatively,
the
ideas
exposed
in
the
present
§3.11,
(iv),
may
be
summarized/expressed
in
terms
of
elementary
numerical
manipulations,
as
follows.
First
of
all,
the
overall
general
logical
flow
of
inter-universal
Teichmüller
theory
—
i.e.,
starting
from
the
definition
of
the
Θ-link,
proceeding
to
the
multiradial
representation
of
the
Θ-pilot
[cf.
[IUTchIII],
Theorem
3.11],
and
finally,
culminating
in
a
final
numerical
estimate
[cf.
[IUTchIII],
Corollary
3.12]
—
may
be
represented
by
means
of
real
numbers
A,
B
∈
R
>0
and
,
N
∈
R
such
that
0
≤
<
1
in
the
following
way:
·
Θ-link:
def
N
=
−2B
∧
def
N
=
−A
;
·
multiradial
representation
of
the
Θ-pilot:
N
=
−2A
+
∧
N
=
−A
;
·
final
numerical
estimate:
−2A
+
=
−A,
hence
A
=
,
i.e.,
A
<
1.
and
and
Thus,
the
Θ-link
[cf.
(1
)]
and
multiradial
representation
of
the
Θ-pilot
[cf.
(2
)]
are
meaningful/nontrivial
precisely
because
of
the
logical
relator
“∧”,
whose
use
obligates
one,
in
the
definition
of
the
Θ-link,
to
consider
a
priori
distinct
real
numbers
A,
B;
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
143
the
passage
from
the
multiradial
representation
of
the
Θ-pilot
to
the
final
numerical
and
estimate
is
immediate/straightforward/logically
transparent
[cf.
(3
)].
By
contrast,
if
one
replaces
“∧”
by
“∨”,
then
our
elementary
numerical
model
of
the
logical
structure
of
inter-universal
Teichmüller
theory
takes
the
following
form:
·
“∨”
version
of
Θ-link:
def
N
=
−2B
∨
def
N
=
−A
[cf.
def
N
=
−2A
∨
def
N
=
−A
];
·
“∨”
version
of
multiradial
representation
of
the
Θ-pilot:
N
=
−2A
+
∨
N
=
−A
;
·
final
numerical
estimate:
−2A
+
=
−A,
hence
A
=
,
i.e.,
A
<
1.
That
is
to
say,
the
use
of
distinct
real
numbers
A,
B
in
the
definition
of
the
“∨”
version
or
of
Θ-link
seems
entirely
superfluous
[cf.
(1
)].
This
motivates
one
to
identify
A
and
B
—
i.e.,
to
suppose
“for
the
sake
of
simplicity”
that
A
=
B
—
which
then
has
the
effect
of
rendering
the
definition
of
the
original
“∧”
version
of
the
Θ-link
invalid/self-
and
or
contradictory
[cf.
(1
),
(1
)].
Once
one
identifies
A
and
B,
i.e.,
once
one
supposes
“for
the
sake
of
simplicity”
that
A
=
B,
the
passage
from
the
“∨”
version
of
Θ-link
to
the
[resulting
“∨”
version
of
the]
multiradial
representation
of
the
Θ-pilot
seems
or
entirely
meaningless/devoid
of
any
interesting
content
[cf.
(2
)].
The
passage
from
the
[resulting
meaningless
“∨”
version
of
the]
multiradial
representation
of
the
Θ-pilot
to
the
final
numerical
estimate
then
seems
abrupt/mysterious/entirely
unjustified,
i.e.,
put
another
way,
looks
as
if
one
erroneously
replaced
the
“∨”
in
the
meaningless
“∨”
version
of
the
mul-
tiradial
representation
of
the
Θ-pilot
by
an
“∧”
without
any
mathematical
or
justification
whatsoever
[cf.
(3
)].
This
is
precisely
the
pernicious
chain
of
misunderstandings
that
has
given
rise
to
a
substantial
amount
of
unnecessary
confusion
concerning
inter-universal
Teichmüller
theory.
(v)
Closed
loops
and
two-dimensionality:
In
the
context
of
the
discussion
of
(iv),
it
is
useful
to
observe
that
one
way
to
think
of
the
construction
algorithm
of
the
multiradial
representation
of
the
Θ-pilot
is
as
follows
[cf.
the
discussion
of
[IUTchIII],
Remark
3.9.5,
(viii)]:
Shinichi
Mochizuki
144
(1
cld
(2
cld
)
This
construction
may
be
thought
of
as
a
construction
of
a
certain
subquotient
of
the
portion
of
the
log-theta-lattice
on
the
right-hand
side
of
Fig.
3.6
[i.e.,
consisting
of
two
vertical
lines
of
log-links
joined
by
a
single
horizontal
Θ-link]
that
restricts
to
the
identity
on
the
vertical
line
of
log-links
that
contains
the
codomain
of
the
Θ-link.
)
Alternatively,
this
subquotient
may
be
thought
of
as
a
sort
of
projection
of
the
the
“Θ-intertwining”
[i.e.,
the
structure
on
an
abstract
F
×μ
-prime-
strip
as
the
F
×μ
-prime-strip
arising
from
the
Θ-pilot
object
appearing
in
the
domain
of
the
Θ-link]
—
up
to
suitable
indeterminacies
—
onto
the
“q-intertwining”
[i.e.,
the
structure
on
an
abstract
F
×μ
-prime-
strip
as
the
F
×μ
-prime-strip
arising
from
the
q-pilot
object
appearing
in
the
codomain
of
the
Θ-link].
Here,
we
observe
further
[cf.
the
discussion
of
§3.10,
(ii)]
that
this
algorithm
converts
any
collection
of
input
data
—
i.e.,
any
F
×μ
-prime-strip
—
into
output
data
that
is
expressed
in
terms
of
the
arithmetic
holomorphic
structure
of
the
input
data
F
×μ
-prime-strip,
hence
makes
it
possible
to
construct,
in
effect,
(3
)
a
single
F
×μ
-prime-strip
that
is
simultaneously
equipped
with
both
the
q-intertwining
and
the
Θ-intertwining,
regarded
up
to
suitable
indeterminacies
[cf.
discussion
of
the
crucial
logical
relator
“∧”
in
(iv)
above;
the
discussion
of
the
final
portion
of
[IUTchIII],
Remark
3.9.5,
(ix)].
cld
Put
another
way,
this
algorithm
allows
one
to
(4
)
construct
a
closed
loop
—
i.e.,
from
a
given
input
data
[q-intertwined!]
F
×μ
-
prime-strip
back
to
the
given
input
data
[q-intertwined!]
F
×μ
-prime-strip
—
whose
output
data
consists
of
the
Θ-intertwining
[up
to
suitable
indeterminacies]
on
this
single
given
input
data
[q-intertwined!]
F
×μ
-prime-strip
[cf.
the
discussion
of
the
final
portion
of
[IUTchIII],
Remark
3.9.5,
(ix)].
cld
Here,
we
note
that
it
is
precisely
this
closed
nature
of
the
loop
that
allows
one
to
est
est
est
derive
[cf.
§3.7,
(ii),
(10
),
(11
),
(12
)]
nontrivial
consequences
[cf.
the
discussion
and
or
of
(3
),
(3
)
in
(iv)]
from
the
multiradial
representation
of
the
Θ-pilot.
That
is
to
say,
one
entirely
elementary/“general
nonsense”
observation
that
may
be
made
in
this
context
is
the
following:
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
(5
145
cld
)
If,
by
contrast,
the
algorithm
only
yielded
[not
a
closed
loop,
but
rather]
an
“open
path”
—
i.e.,
from
one
[“input”]
type
of
mathematical
object
to
some
distinct/non-
comparable
[“output”]
type
of
mathematical
object
—
then
one
could
only
conclude
from
the
algorithm
some
sort
of
relationship
between
the
structure
of
the
input
object
and
the
structure
of
the
output
object;
it
would
not,
however,
be
possible
to
conclude
anything
about
the
intrinsic
structure
of
either
the
input
or
the
output
objects.
In
this
context,
it
is
also
important
to
note
the
crucial
role
played
by
the
notion/definition
[cf.
[IUTchII],
Definition
4.9,
(viii)]
of
an
“F
×μ
-prime-strip”,
i.e.,
by
the
particular
sort
of
data
that
appears
in
an
F
×μ
-prime-strip.
That
is
to
say:
(6
)
The
data
contained
in
an
F
×μ
-prime-strip
is,
on
the
one
hand,
sufficiently
strong
to
suffice
as
input
data
for
the
construction
algorithm
of
the
multiradial
representation
of
the
Θ-pilot,
but,
on
the
other
hand,
sufficiently
weak
so
as
to
yield
isomorphic
collections
of
data
[hence,
in
particular,
to
allow
one
to
define
the
Θ-link!]
from
the
data
arising
from
the
q-pilot
and
Θ-pilot
objects.
cld
Indeed,
it
is
precisely
this
simultaneous
sufficient
strength/weakness
that
makes
it
possible
to
construct
a
single
F
×μ
-prime-strip
that
is
simultaneously
equipped
with
both
the
q-intertwining
and
the
Θ-intertwining,
regarded
up
to
suitable
inde-
cld
terminacies
[cf.
(3
)].
Here,
we
note
that,
at
a
more
concrete
level:
(7
cld
cld
)
The
crucial
sufficient
strength/weakness
properties
discussed
in
(6
)
may
be
understood
as
a
consequence
of
the
fact
that
an
F
×μ
-prime-strip
is
comprised
of
both
unit
group
and
value
group
portions
—
i.e.,
of
portions
corresponding
to
the
“two
arithmetic/combinatorial
dimensions”
of
the
discussion
of
§2.11
—
but
comprised
in
such
a
way
that
these
two
arithmetic/combinatorial
dimensions
are
independent
of
one
another,
i.e.,
not
[at
least
in
any
a
priori
sense!]
subject
to
any
ring
structure/intertwining
such
as
the
q-
or
Θ-intertwinings.
Moreover,
as
discussed
in
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob7);
[IUTchIII],
Remark
3.9.5,
(ix),
(x):
(8
cld
cld
cld
)
The
crucial
sufficient
strength/weakness
properties
discussed
in
(6
),
(7
)
would
fail
to
hold
if
various
portions
of
the
collection
of
data
that
constitutes
an
F
×μ
-
prime-strip
are
omitted.
For
instance
[cf.
the
discussion
of
[IUTchIII],
Remark
3.9.5,
(vii),
(Ob7);
[IUTchIII],
Remark
3.9.5,
(ix),
(x)]:
(9
cld
)
If,
in
the
argument
of
§3.7,
(ii),
one
omits
the
formation
of
the
holomorphic
est
hull
[cf.
§3.7,
(ii),
(8
)],
then
the
resulting
argument
amounts,
in
essence,
to
Shinichi
Mochizuki
146
cld
cld
an
attempt
to
establish
a
closed
loop
as
in
(3
),
(4
),
in
a
situation
in
which
F
×μ
-prime-strips
are
replaced
by
some
alternative
type
of
mathematical
object
cld
that
does
not
satisfy
[cf.
(8
)]
the
crucial
sufficient
strength/weakness
properties
cld
cld
discussed
in
(6
),
(7
),
hence
does
not
give
rise
to
such
a
closed
loop
or,
in
cld
particular,
to
the
nontrivial
conclusions
arising
from
a
closed
loop
[cf.
(5
)].
§
4.
§
4.1.
Historical
comparisons
and
analogies
Numerous
connections
to
classical
theories
Many
discussions
of
inter-universal
Teichmüller
theory
exhibit
a
tendency
to
em-
phasize
the
novelty
of
many
of
the
ideas
and
notions
that
constitute
the
theory.
On
the
other
hand,
another
important
aspect
of
many
of
these
ideas
and
notions
of
inter-
universal
Teichmüller
theory
is
their
quite
substantial
relationship
to
numerous
classical
theories.
One
notable
consequence
of
this
latter
aspect
of
inter-universal
Teichmüller
theory
is
the
following:
one
obstacle
that
often
hampers
the
progress
of
mathematicians
in
their
study
of
inter-universal
Teichmüller
theory
is
a
lack
of
familiarity
with
such
classical
theories,
many
of
which
date
back
to
the
1960’s
or
1970’s
[or
even
earlier]!
(i)
Contrast
with
classical
numerical
computations:
sion
by
recalling
We
begin
our
discus-
the
famous
computation
in
the
late
nineteenth
century
by
William
Shanks
of
π
to
707
places,
which
was
later
found,
with
the
advent
of
digital
computing
devices
in
the
twentieth
century,
to
be
correct
only
up
to
527
places!
The
work
that
went
into
this
sort
of
computation
may
strike
some
mathematicians
as
being
reminiscent,
in
a
certain
sense,
of
the
sheer
number
of
pages
of
the
various
papers
—
i.e.,
such
as
[Semi],
[FrdI],
[FrdII],
[EtTh],
[GenEll],
[AbsTopIII],
[IUTchI],
[IUTchII],
[IUTchIII],
[IUTchIV]
—
that
one
must
study
in
order
to
achieve
a
thorough
understanding
of
inter-universal
Teichmüller
theory.
In
fact,
however,
inter-universal
Teichmüller
theory
differs
quite
fundamentally
from
the
computation
of
Shanks
in
that,
as
was
discussed
throughout
the
present
paper,
and
especially
in
§3.8,
§3.9,
§3.10,
§3.11,
the
central
ideas
of
inter-universal
Teichmüller
theory
are
rather
compact
and
con-
ceptual
in
nature
and
revolve
around
the
issue
of
comparison,
by
means
of
the
notions
of
mono-anabelian
transport
and
multiradiality,
of
mutually
alien
copies
of
miniature
models
of
conven-
tional
scheme
theory
in
a
fashion
that
exhibits
remarkable
similarities
to
the
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
147
compact
and
conceptual
nature
of
the
classical
computation
of
the
Gaussian
integral
by
means
of
the
introduction
of
two
“mutually
alien
copies”
of
this
integral.
One
way
of
briefly
summarizing
these
remarkable
similarities
[i.e.,
which
are
discussed
in
more
detail
in
§3.8,
as
well
as
in
the
Introduction
to
the
present
paper]
is
as
follows:
·
the
“theta
portion”
—
i.e.,
the
Θ-link
—
of
the
log-theta-lattice
of
inter-
universal
Teichmüller
theory
may
be
thought
of
as
a
sort
of
statement
of
the
main
computational
problem
of
inter-universal
Teichmüller
theory
and
may
be
understood
as
corresponding
to
the
various
Gaussians
that
appear
in
the
classical
computation
of
the
Gaussian
integral,
while
·
the
“log
portion”
—
i.e.,
the
log-link
—
of
the
log-theta-lattice
of
inter-
universal
Teichmüller
theory
may
be
thought
of
as
a
sort
of
solution
of
the
main
computational
problem
of
inter-universal
Teichmüller
theory
and
may
be
understood
as
corresponding
to
the
angular
portion
of
the
representation
via
polar
coordinates
of
the
[square
of
the]
Gaussian
integral.
In
this
context,
it
is
of
interest
to
recall
the
remarkable
similarities
of
certain
aspects
of
inter-universal
Teichmüller
theory
to
the
theory
surrounding
the
functional
equation
of
the
theta
function
—
i.e.,
“Jacobi’s
identity”
[cf.
the
discussion
of
the
final
portion
of
[Pano],
§3;
the
discussion
preceding
[Pano],
Theorem
4.1]
—
which
may
be
thought
of
as
a
sort
of
function-theoretic
version
of
the
computation
of
the
Gaussian
integral
that
may
be
obtained,
roughly
speaking,
by
interpreting
this
computation
of
the
Gaussian
integral
in
the
context
of
the
hyperbolic
geometry
of
the
upper
half-
plane.
As
discussed
in
§2.4;
Example
3.2.1,
(iii);
§3.3,
(v),
the
analogy
between
certain
aspects
of
inter-universal
Teichmüller
theory
and
the
hyperbolic
geometry
of
the
upper
half-plane
has
not
been
exposed
in
the
present
paper
in
much
detail
since
it
has
already
been
exposed
in
substantial
detail
in
[BogIUT]
[cf.
also
§3.10,
(vi);
§4.3,
(iii),
of
the
present
paper].
On
the
other
hand,
it
is
of
interest
to
recall
that
classically,
Jacobi’s
identity
was
often
appreciated
as
a
relatively
compact
and
conceptual
way
to
achieve
a
startling
improvement
in
computational
accuracy
in
the
context
of
explicit
numerical
calculations
of
values
of
the
theta
function
[cf.
the
discussion
preceding
[Pano],
Theorem
4.1].
(ii)
Explicit
examples
of
connections
to
classical
theories:
Next,
we
review
various
explicit
examples
of
connections
between
inter-universal
Teichmüller
theory,
as
exposed
thus
far
in
the
present
paper,
and
various
classical
theories:
cls
(1
)
Recall
from
the
discussion
of
§2.10
that
the
notion
of
a
“universe”,
as
well
as
the
use
of
multiple
universes
within
the
discussion
of
a
single
set-up
in
arithmetic
148
Shinichi
Mochizuki
geometry,
already
occurs
in
the
mathematics
of
the
1960’s,
i.e.,
in
the
mathematics
of
Galois
categories
and
étale
topoi
associated
to
schemes
[cf.
[SGA1],
[SGA4]].
cls
(2
)
One
important
aspect
of
the
appearance
of
universes
in
the
theory
of
Galois
categories
is
the
inner
automorphism
indeterminacies
that
occur
when
one
relates
Galois
categories
associated
to
distinct
schemes
via
a
morphism
between
such
schemes
[cf.
[SGA1],
Exposé
V,
§5,
§6,
§7].
These
indeterminacies
may
be
regarded
as
distant
ancestors,
or
prototypes,
of
the
more
drastic
indeterminacies
—
cf.,
e.g.,
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
discussed
in
§3.7,
(i)
—
that
occur
in
inter-universal
Teichmüller
theory.
cls
(3
)
The
theory
of
Tate
developed
in
the
1960’s
[cf.
[Serre],
Chapter
III,
Appendix]
concerning
Hodge-Tate
representations
plays
a
fundamental
role
in
the
theory
of
[Q
p
GC]
[cf.
also
[AbsTopI],
§3].
This
theory
of
[Q
p
GC]
and
[AbsTopI],
§3,
may
be
regarded
as
a
precursor
of
the
theory
of
log-shells
developed
in
[AbsTopIII],
§3,
§4,
§5.
cls
(4
)
The
approach
of
Faltings
to
p-adic
Hodge
theory
via
the
technique
of
almost
étale
extensions
[cf.
[Falt2]]
plays
a
central
role
in
the
p-adic
anabelian
geometry
developed
in
[pGC].
Here,
we
recall
that
this
theory
of
[pGC]
constitutes
the
crucial
technical
tool
that
underlies
the
Belyi
and
elliptic
cuspidalizations
of
[AbsTopII],
§3,
which,
in
turn,
play
a
quite
essential
role
in
the
theory
of
[EtTh],
[AbsTopIII],
hence,
in
particular,
in
inter-universal
Teichmüller
theory.
cls
(5
)
The
work
of
Tate
in
the
1960’s
concerning
theta
functions
on
uniformizations
of
Tate
curves
[cf.
[Mumf2],
§5]
plays
a
fundamental
role
in
[EtTh],
hence
also
in
inter-universal
Teichmüller
theory.
cls
(6
)
The
scheme-theoretic
Hodge-Arakelov
theory
discussed
in
Example
2.14.3
and
§3.9
may
be
regarded
as
a
natural
extension
of
[the
portion
concerning
elliptic
curves
of]
Mumford’s
theory
of
algebraic
theta
functions
[cf.
[Mumf1]].
cls
(7
)
The
invariance
of
the
étale
site,
up
to
isomorphism,
with
respect
to
the
Frobe-
nius
morphism
in
positive
characteristic
[cf.
the
discussion
of
Example
2.6.1,
(i)]
was
well-known
[cf.
[SGA1],
Exposé
IX,
Théorème
4.10]
to
the
Grothendieck
school
in
the
1960’s.
As
discussed
in
§2.6,
§2.7,
this
phenomenon,
taken
together
with
the
fundamental
work
of
Uchida
[cf.
[Uchi]]
in
the
1970’s
concerning
the
anabelian
geometry
of
one-dimensional
function
fields
over
a
finite
field,
may
be
regarded
as
the
fundamental
prototype
for
the
apparatus
of
mono-anabelian
transport
—
and,
in
particular,
for
the
terms
“Frobenius-like”
and
“étale-like”
—
which
plays
a
central
role
in
inter-universal
Teichmüller
theory.
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
149
cls
(8
)
The
use
of
abstract
[commutative]
monoids,
e.g.,
in
the
theory
of
Frobe-
nioids
[cf.
§3.3,
(iii);
§3.5],
which
plays
a
fundamental
role
in
inter-universal
Teichmüller
theory,
was
motivated
by
the
use
of
such
monoids
in
the
theory
of
log
schemes
[cf.
[Kato1],
[Kato2]],
which,
in
turn,
was
motivated
by
the
use
of
such
monoids
in
the
classical
theory
of
toric
varieties
developed
in
the
1970’s
[cf.
[KKMS]].
cls
(9
)
Recall
from
the
discussion
of
§3.1,
(iv),
(v),
that
the
notion
of
multiradial-
ity,
which
plays
a
fundamental
role
in
inter-universal
Teichmüller
theory,
may
be
regarded
as
a
sort
of
abstract
combinatorial
analogue
of
the
Grothendieck
defini-
tion
of
a
connection,
i.e.,
which
plays
a
central
role
in
the
classical
theory
of
the
crystalline
site
and
dates
back
to
the
1960’s
[cf.
[GrCrs]].
cls
(iii)
Monoids
and
Galois
theory:
With
regard
to
(ii),
(8
),
we
observe
that
the
important
role
played
by
abstract
[commutative]
monoids
in
inter-universal
Te-
ichmüller
theory
is
reminiscent
of
the
way
in
which
such
monoids
are
used
by
many
mathematicians
in
research
related
to
“geometry
over
F
1
”
[i.e.,
the
fictitious
“field
with
one
element”].
On
the
other
hand,
the
way
in
which
such
monoids
are
used
in
inter-universal
Teichmüller
theory
differs
fundamentally
from
the
way
in
which
such
monoids
are
used
in
conventional
research
on
geometry
over
F
1
in
the
following
respect:
·
in
inter-universal
Teichmüller
theory,
various
anabelian
and
Kummer-
theoretic
aspects
of
Galois
or
arithmetic
fundamental
groups
that
act
on
such
monoids
play
a
fundamental
role
[cf.
the
discussion
of
mono-anabelian
transport
in
§2.7,
§2.9!];
·
by
contrast,
at
least
to
the
author’s
knowledge
at
the
time
of
writing,
research
on
geometry
over
F
1
does
not
involve,
in
any
sort
of
essential
way,
such
anabelian
or
Kummer-theoretic
aspects
of
Galois
or
arithmetic
fundamental
groups
acting
on
monoids.
Indeed,
this
fundamental
difference
between
inter-universal
Teichmüller
theory
and
con-
ventional
research
on
geometry
over
F
1
might
give
rise
to
various
interesting
questions
and
hence
stimulate
further
research.
Finally,
in
this
context,
it
is
perhaps
of
interest
to
note
that
although
there
is
no
specific
mathematical
object
in
inter-universal
Teichmüller
theory
that
may
be
said
to
correspond
to
“F
1
”,
in
some
sense
the
notion
of
a
“field
with
one
element”
may,
at
a
more
conceptual
level,
be
thought
of
as
corresponding
to
the
notion
of
coricity/cores/coric
objects
—
that
is
to
say,
objects
that
are
invariant
with
respect
to
[i.e.,
“lie
under,
in
a
unique
way”]
various
operations
such
as
links
[cf.
the
discussion
of
§2.7,
(iv)]
—
a
notion
which
is
indeed
central
to
inter-universal
Teichmüller
theory.
150
Shinichi
Mochizuki
(iv)
Techniques
to
avoid
stacks
and
2-categories:
Some
mathematicians
ap-
pear
to
have
a
strong
aversion
to
the
use
of
such
notions
as
“categories
of
categories”
or
algebraic
stacks
—
i.e.,
notions
that
obligate
one
to
work
with
2-categories
—
in
arithmetic
geometry.
Here,
we
observe
that
the
substantive
mathematical
phenomenon
that
obligates
one,
in
such
situations,
to
work
with
2-categories
is
essentially
the
same
phenomenon
as
the
phenomenon
constituted
by
the
inner
automorphism
indeter-
cls
minacies
discussed
in
(ii),
(2
).
On
the
other
hand,
in
inter-universal
Teichmüller
theory,
various
“general
nonsense”
techniques
are
applied
that
allow
one,
in
such
sit-
uations,
to
work
with
categories
[i.e.,
as
opposed
to
2-categories]
and
thus
avoid
the
cumbersome
complications
that
arise
from
working
with
2-categories:
·
In
inter-universal
Teichmüller
theory,
one
typically
works
with
slim
cat-
egories
such
as
Galois
categories
that
arise
from
slim
profinite
groups
[i.e.,
profinite
groups
for
which
the
centralizer
of
every
open
subgroup
is
trivial]
or
temp-slim
tempered
groups
[cf.
[Semi],
Remark
3.4.1].
The
use
of
slim
cate-
gories
allows
one,
in
effect,
to
think
of
“categories
of
categories”
as
categories
[i.e.,
rather
than
2-categories].
Indeed,
in
the
case
of
slim
profinite
groups,
this
point
of
view
is
precisely
the
point
of
view
that
underlies
the
theory
of
[GeoAnbd].
Generalities
concerning
slim
categories
may
be
found
in
[FrdI],
Ap-
pendix.
·
Another
important
“general
nonsense”
technique
that
is
used
in
inter-universal
Teichmüller
theory
to
keep
track
explicitly
of
the
various
types
of
indetermi-
nacies
that
occur
is
the
notion
of
a
poly-morphism
[cf.
[IUTchI],
§0],
i.e.,
a
[possibly
empty]
subset
of
the
set
of
arrows
between
two
objects
in
a
category.
Thus,
there
is
a
natural
way
to
compose
two
poly-morphisms
[i.e.,
that
consist
of
composable
arrows]
to
obtain
a
new
poly-morphism.
Consideration
of
such
composites
of
poly-morphisms
allows
one
to
trace
how
various
indeterminacies
interact
with
one
another.
In
this
context,
it
is
perhaps
useful
to
observe
that,
from
a
more
classical
point
of
view,
cls
the
inner
automorphism
indeterminacies
discussed
in
(ii),
(2
),
corre-
spond
to
the
indeterminacy
in
the
choice
of
a
basepoint
of
a
[say,
connected,
locally
contractible]
topological
space.
That
is
to
say,
in
anabelian
geometry,
working
with
slim
anabelioids
as
opposed
to
slim
profinite
groups
corresponds,
in
essence,
to
working,
in
classical
topology,
with
topological
spaces
as
opposed
to
pointed
topological
spaces.
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
151
Since
many
natural
maps
between
topological
spaces
—
i.e.,
such
as
localization
maps!
—
are
not
compatible
with
choices
of
distinguished
points,
it
is
often
more
natural,
in
many
discussions
of
classical
topology,
to
make
use
[not
only
of
the
notion
of
a
“pointed
topological
space”,
but
also]
of
the
notion
of
a
“topological
space”
[i.e.,
that
is
not
equipped
with
the
choice
of
a
distinguished
point!].
It
is
precisely
for
this
reason
that
in
many
discussions
—
i.e.,
such
as
those
that
occur
in
inter-universal
Teichmüller
theory,
for
instance,
in
the
case
of
localizations
at
various
primes
of
an
NF!
[cf.
the
discussion
of
§3.3,
(iv),
(v),
(vi)]
—
involving
the
geometry
of
categories,
it
is
much
more
natural
and
less
cumbersome
to
work
with
slim
categories
such
as
slim
anabelioids
[i.e.,
as
opposed
to
profinite
groups].
(v)
Notational
complexity
and
mutually
alien
copies:
Some
readers
of
the
papers
[IUTchI],
[IUTchII],
[IUTchIII],
[IUTchIV]
have
expressed
bafflement
at
the
de-
gree
of
complexity
of
the
notation
—
i.e.,
by
comparison
to
the
degree
of
complexity
of
notation
that
is
typical
in
conventional
papers
on
arithmetic
geometry
—
that
ap-
pears
in
these
papers.
This
complexity
of
notation
may
be
understood
as
a
natural
consequence
of
·
the
need
to
distinguish
between
objects
that
belong
to
distinct
copies,
i.e.,
distinct
“miniature
models”,
of
conventional
scheme
theory
[cf.,
e.g.,
the
labels
“n,
m”
for
the
various
lattice
points
“•”
in
the
log-theta-lattice,
as
discussed
in
Fr/ét
)];
§3.3,
(ii);
§3.6,
(iv);
[IUTchIII],
Definition
3.8,
(iii)],
together
§2.8,
(2
with
·
the
need
to
distinguish
between
distinct
objects
—
such
as
distinct
cyclo-
tomes
related
by
nontrivial
cyclotomic
rigidity
isomorphisms
[cf.,
e.g.,
the
discussion
of
§2.6,
§2.12,
§2.13,
§3.4,
as
well
as
the
discussion
of
§4.2,
(i),
be-
low]
—
within
a
single
miniature
model
of
conventional
scheme
theory
that
are
related
to
one
another
via
structures
that
are
“taken
for
granted”
in
conven-
tional
discussions
of
arithmetic
geometry,
but
whose
precise
specification
is
in
fact
highly
nontrivial
in
the
context
of
situations
where
one
considers
multiple
miniature
models
of
conventional
scheme
theory.
Put
another
way,
this
complexity
of
notation
may
be
regarded
as
an
inevitable
conse-
quence
of
the
central
role
played
in
inter-universal
Teichmüller
theory
by
“mutually
alien
copies/multiple
miniature
models”
of
conventional
scheme
theory
and
the
resulting
inter-universality
issues
that
arise
[cf.
the
discussion
of
§2.7,
(i),
(ii);
§2.10;
§3.8].
In
particular,
this
complexity
of
notation
is
by
no
means
superfluous.
§
4.2.
Contrasting
aspects
of
class
field
theory
and
Kummer
theory
We
begin
our
discussion
by
observing
that
the
role
played
by
local
class
field
theory
[cf.
the
discussion
of
§2.11;
§2.12,
especially
Example
2.12.1,
(ii),
(iii);
§3.4,
(v)]
152
Shinichi
Mochizuki
in
inter-universal
Teichmüller
theory
is,
in
many
respects,
not
particularly
prominent,
while
global
class
field
theory
[for
NF’s]
is
entirely
absent
from
inter-universal
Teichmüller
theory.
This
situation
for
class
field
theory
contrasts
sharply
with
the
very
central
role
played
by
Kummer
theory
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
mono-anabelian
transport
in
§2.7,
§2.9!].
In
fact,
this
state
of
affairs
is
both
natural
and
indeed
somewhat
inevitable
for
a
number
of
reasons,
which
we
pause
to
survey
in
the
discussion
to
follow
[cf.
also
Fig.
4.1
below].
(i)
Strong
functoriality
properties
and
the
central
role
of
cyclotomic
rigidity
in
Kummer
theory:
Perhaps
the
most
conspicuous
difference
between
class
field
theory
and
Kummer
theory
is
the
fact
that,
whereas
·
class
field
theory
may
only
be
formulated
for
a
certain
special
class
of
arithmetic
fields
[a
class
which
in
fact
includes
the
function
fields
of
all
the
integral
schemes
that
appear
in
inter-universal
Teichmüller
theory],
e.g.,
for
global
fields
[i.e.,
fields
that
are
finitely
generated
over
an
NF
or
a
finite
field]
or
certain
types
of
completions
or
localizations
of
such
global
fields,
·
Kummer
theory,
by
contrast,
may
be
formulated,
by
using
the
Kummer
exact
sequence
in
étale
cohomology
[cf.,
e.g.,
the
discussion
at
the
beginning
of
[Cusp],
§2],
for
[essentially]
arbitrary
types
of
schemes
and
even
for
abstract
monoids
[cf.
[FrdII],
Definition
2.1,
(ii)]
that
satisfy
relatively
weak
conditions
and
do
not
necessarily
arise
from
the
multiplicative
structure
of
a
commutative
ring.
A
closely
related
difference
between
class
field
theory
and
Kummer
theory
is
the
fact
that,
whereas
·
class
field
theory
only
satisfies
very
limited
functoriality
properties,
i.e.,
for
finite
separable
field
extensions
and
certain
types
of
localization
operations
associated
to
a
valuation,
·
Kummer
theory
satisfies
very
strong
functoriality
properties,
for
[essen-
tially]
arbitrary
morphisms
between
[essentially]
arbitrary
schemes
or
between
abstract
monoids
that
satisfy
suitable,
relatively
weak
conditions.
These
properties
of
Kummer
theory
make
Kummer
theory
much
more
suitable
for
use
in
anabelian
geometry,
where
it
is
natural
to
consider
morphisms
between
arithmetic
fundamental
groups
that
correspond
to
quite
general
morphisms
between
quite
general
schemes,
i.e.,
where
by
“quite
general”,
we
mean
by
comparison
to
the
restrictions
that
arise
if
one
attempts
to
apply
class
field
theory.
Perhaps
the
most
fundamental
example
of
this
sort
of
situation
[i.e.,
that
is
of
interest
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
153
in
anabelian
geometry,
but
to
which
class
field
theory
cannot,
at
least
in
any
immediate
way,
be
applied]
is
the
situation
that
arises
if
one
considers
the
operation
of
evaluation
of
various
types
of
functions
on,
say,
a
curve,
at
a
closed
point
of
the
curve
[cf.
the
discussion
of
(ii)
below;
Example
2.13.1,
(iv);
§2.14;
§3.6;
[IUTchIV],
Remark
2.3.3,
(vi)].
On
the
other
hand,
one
highly
nontrivial
and
quite
delicate
aspect
of
Kummer
theory
that
does
not
appear
in
class
field
theory
is
the
issue
of
establishing
cyclotomic
rigidity
isomorphisms
between
cyclotomes
con-
structed
from
the
various
rings,
monoids,
Galois
groups,
or
arithmetic
fun-
damental
groups
that
appear
in
a
particular
situation.
Various
examples
of
such
isomorphisms
between
cyclotomes
may
be
seen
in
the
theory
discussed
in
[PrfGC],
[the
discussion
preceding]
Lemma
9.1;
[AbsAnab],
Lemma
2.5;
[Cusp],
Proposition
1.2,
(ii);
[FrdII],
Theorem
2.4,
(ii);
[EtTh],
Corollary
2.19,
(i);
[AbsTopIII],
Corollary
1.10,
(ii);
[AbsTopIII],
Remarks
3.2.1,
3.2.2
[cf.
also
Example
2.12.1,
(ii);
Example
2.13.1,
(ii);
§3.4,
(ii),
(iii),
(iv),
(v),
of
the
present
paper].
All
of
these
examples
concern
“Kummer-faithful”
situations
[cf.
[AbsTopIII],
Definition
1.5],
i.e.,
situations
in
which
the
Kummer
map
on
the
multiplicative
monoid
[e.g.,
which
arises
from
the
multiplicative
structure
of
a
ring]
of
interest
is
injective.
Then
the
cyclotomic
rigidity
issues
that
arise
typically
involve
the
cyclotomes
obtained
by
considering
the
torsion
subgroups
of
such
multiplicative
monoids.
This
sort
of
situ-
ation
contrasts
sharply
with
the
sort
of
highly
“non-Kummer-faithful”
situation
con-
sidered
in
[PopBog],
i.e.,
where
one
works
with
function
fields
over
algebraic
closures
of
finite
fields
[cf.
[PopBog],
Theorem
I].
That
is
to
say,
in
the
sort
of
situation
considered
in
[PopBog],
the
Kummer
map
vanishes
on
the
roots
of
unity
of
the
base
field,
and
the
Kummer
theory
that
is
applied
[cf.
[PopBog],
§5.2]
does
not
revolve
around
the
issue
of
establishing
cyclotomic
rigidity
isomorphisms.
In
particular,
in
the
context
of
this
sort
of
application
of
Kummer
theory,
it
is
natural
to
think
of
the
image
of
the
Kummer
map
as
a
sort
of
projective
space,
i.e.,
a
quotient
by
the
action
of
multiplication
by
nonzero
elements
of
the
base
field.
Thus,
in
summary,
the
relationship
just
discussed
between
“Kummer-faithful
Kummer
theory”
and
“non-Kummer-faithful
Kummer
theory”
may
be
thought
of
as
the
difference
between
“
injective
Kummer
theory”
and
“
projective
Kummer
theory”.
(ii)
The
functoriality
of
Kummer
theory
with
respect
to
evaluation
of
special
functions
at
torsion
points:
As
mentioned
in
(i),
the
operation
of
eval-
uation
of
various
types
of
[special]
functions
on,
say,
a
curve,
at
various
types
of
154
Shinichi
Mochizuki
[special]
points
of
the
curve
plays
a
fundamental
role
in
the
Kummer
theory
that
is
applied
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
Example
2.13.1,
(iv);
§2.14;
§3.6;
[IUTchIV],
Remark
2.3.3,
(vi)].
This
contrasts
sharply
with
the
fact
that
class
field
theory
may
only
be
related
to
the
operation
of
evaluation
of
special
functions
at
special
points
in
very
restricted
classical
cases,
namely,
the
theory
of
exponential
functions
in
the
case
of
Q
or
modular
and
elliptic
functions
in
the
case
of
imaginary
quadatic
fields.
Indeed,
the
goal
of
generalizing
the
theory
that
exists
in
these
very
restricted
cases
to
the
case
of
arbitrary
NF’s
is
precisely
the
content
of
Kronecker’s
Jugendtraum,
or
Hilbert’s
twelfth
problem
[cf.
the
discussion
of
[IUTchIV],
Remark
2.3.3,
(vii)].
Indeed,
in
light
of
this
state
of
affairs,
one
is
tempted
to
regard
inter-universal
Teichmüller
theory
as
a
sort
of
“realization/solution”
of
the
“version”
of
Kronecker’s
Jugendtraum,
or
Hilbert’s
twelfth
problem,
that
one
obtains
if
one
replaces
class
field
theory
by
Kummer
theory!
(iii)
The
arithmetic
holomorphicity
of
global
class
field
theory
versus
the
mono-analyticity
of
Kummer
theory:
Another
important
aspect
of
the
fun-
damental
differences
between
class
field
theory
and
Kummer
theory
that
were
highlighted
in
the
discussion
of
(i)
is
the
following:
·
whereas
the
essential
content
of
class
field
theory
reflects
various
delicate
arithmetic
properties
that
are
closed
related
to
the
arithmetic
holomorphic
structure
of
the
very
restricted
types
of
arithmetic
fields
to
which
it
may
be
applied,
·
the
very
general
and
strongly
functorial
nature
of
Kummer
theory
makes
Kummer
theory
more
suited
to
treating
the
sorts
of
mono-analytic
struc-
tures
that
arise
in
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
§2.7,
(vii)].
Indeed,
at
a
very
naive
level,
this
phenomenon
may
be
seen
in
the
difference
between
the
“input
data”
for
class
field
theory
and
Kummer
theory,
i.e.,
very
restricted
arithmetic
fields
in
the
case
of
class
field
theory
versus
very
general
types
of
abstract
multiplicative
monoids
in
the
case
of
Kummer
theory
[cf.
the
discussion
of
(i)].
Another
important
instance
of
this
phenomenon
may
be
seen
in
the
fact
that
whereas
·
the
global
reciprocity
law,
which
plays
a
central
role
in
class
field
the-
ory
for
NF’s,
involves
a
nontrivial
“intertwining”
relationship,
for
any
prime
number
l,
between
the
local
unit
determined
by
l
at
nonarchimedean
valuations
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
Class
field
theory
Kummer
theory
may
be
formulated
only
for
special
arithmetic
fields
may
be
formulated
for
very
general
abstract
multiplicative
monoids
satisfies
only
very
limited
functoriality
properties
satisfies
very
strong
functoriality
properties
limited
range
of
applicability
to
anabelian
geometry
wide
range
of
applicability
to
anabelian
geometry
cyclotomic
rigidity
isomorphisms
are
irrelevant
cyclotomic
rigidity
isomorphisms
play
a
central
role
no
known
compatibility
with
evaluation
of
functions
at
points
compatible
with
evaluation
of
functions
at
points
closely
related
to
the
arithmetic
holomorphic
structure
of
very
restricted
types
of
arithmetic
fields
applicable
to
mono-analytic
structures
such
as
abstract
multiplicative
monoids
incompatible
with
local
unit
group/value
group
decouplings
compatible
with
local
unit
group/
value
group
decouplings
related
to
global
Dirichlet
density
of
primes
naturally
applied
in
conjunction
with
Prime
Number
Theorem
verification,
involving
cyclotomic
extensions,
of
the
global
reciprocity
law
global
cyclotomic
rigidity
algorithms
via
×
Z
=
{1}
Q
>0
Fig.
4.1:
Comparison
between
class
field
theory
and
Kummer
theory
155
Shinichi
Mochizuki
156
of
residue
characteristic
=
l
and
the
nonzero
element
of
the
value
group
deter-
mined
by
l
at
nonarchimedean
valuations
of
residue
characteristic
l,
·
the
compatibility
of
the
Kummer
theory
applied
in
inter-universal
Te-
ichmüller
theory
with
various
“splittings”/“decouplings”
between
the
local
unit
group
and
value
group
portions
of
this
Kummer
theory
plays
a
central
role
in
inter-universal
Teichmüller
theory
gau
—
cf.
the
discussion
of
§3.4;
§3.8,
(8
);
[IUTchIV],
Remark
2.3.3,
(v).
A
closely
related
fact
is
the
fact
that
such
local
unit
group/value
group
splittings
are
incompatible
with
[the
multiplicative
version
of]
Hilbert’s
Theorem
90,
which
plays
a
central
role
in
class
field
theory:
that
is
to
say,
in
the
notation
of
Examples
2.12.1,
2.12.2,
one
×
verifies
immediately
that
whereas
H
1
(G
k
,
k
)
=
0,
H
1
(G
k
,
O
k
×
)
=
0,
H
1
(G
k
,
μ
k
)
=
0,
H
1
(G
k
,
μ
k
·
π
k
Q
)
=
0
×
—
where
we
write
μ
k
·
π
k
Q
⊆
k
for
the
subgroup
of
elements
for
which
some
positive
power
∈
π
k
Z
.
Finally,
we
recall
from
the
discussion
of
[IUTchIV],
Remark
2.3.3,
(i),
(ii),
(iv),
that
a
sort
of
analytic
number
theory
version
of
this
phenomenon
may
be
seen
in
the
fact
that
whereas
·
class
field
theory
is
closely
related
—
especially
if
one
takes
the
point
of
view
of
early
approaches
to
class
field
theory
such
as
the
approach
attributed
to
Weber
—
to
the
“coherent
aggregrations”
of
primes
that
appear
in
discussions
of
the
Dirichlet
density
of
primes,
e.g.,
in
the
context
of
the
Tchebotarev
density
theorem,
·
the
Kummer-theoretic
approach
of
inter-universal
Teichmüller
theory
gives
rise
to
the
multiradial
representation
discussed
in
§3.7,
(i),
which
leads
to
log-volume
estimates
[cf.
the
discussion
of
§3.7,
(ii),
(iv);
the
application
of
[IUTchIV],
Proposition
1.6,
and
[IUTchIV],
Proposition
2.1,
(ii),
in
the
explicit
calculations
of
[IUTchIV],
§1,
§2]
that
involve,
in
an
essential
way,
the
Prime
Number
Theorem,
i.e.,
which,
so
to
speak,
counts
primes
“one
by
one”,
in
effect
“deactivating
the
coherent
aggregrations
of
primes”
that
appear
in
discussions
of
the
Dirichlet
density
of
primes.
(iv)
Global
reciprocity
law
versus
global
cyclotomic
rigidity:
Finally,
we
recall
from
the
discussion
of
[“(b-4)”
in]
[IUTchIV],
Remark
2.3.3,
(i),
(ii),
that
·
the
use
of
cyclotomic
extensions
in
classical
approaches
to
verifying
the
global
reciprocity
law
in
class
field
theory
for
NF’s,
i.e.,
to
verifying
that,
in
effect,
the
reciprocity
map
vanishes
on
idèles
that
arise
from
elements
of
the
NF
under
consideration,
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
157
may
be
thought
of
as
corresponding
to
·
the
approach
taken
in
inter-universal
Teichmüller
theory
to
constructing
cyclotomic
rigidity
isomorphisms
for
the
Kummer
theory
related
to
NF’s
[cf.
§3.4,
(ii),
(v)],
i.e.,
in
effect,
by
applying
the
elementary
fact
that
×
Z
=
{1}
[cf.
the
discussion
of
the
latter
portion
of
[IUTchIII],
Q
>0
Remark
3.12.1,
(iii)].
Indeed,
both
of
these
phenomena
concern
the
fact
that
some
version
of
the
product
formula
—
that
is
to
say,
which,
a
priori
[or
from
a
more
naive,
elementary
point
of
view],
is
only
known
to
hold
for
the
Frobenius-like
multiplicative
monoids
that
arise
from
NF’s
—
in
fact
holds
[i.e.,
in
the
form
of
the
global
reciprocity
law
or
the
elementary
fact
×
Z
=
{1}]
at
the
level
of
[étale-like!]
profinite
Galois
groups.
that
Q
>0
§
4.3.
Arithmetic
and
geometric
versions
of
the
Mordell
Conjecture
(i)
Rough
qualitative
connections
with
Faltings’
proof
of
the
Mordell
Conjecture:
First,
we
begin
by
observing
[cf.
[IUTchIV],
Remark
2.3.3,
(i),
(ii),
for
more
details]
that
there
are
numerous
rough,
qualitative
correspondences
—
some
of
which
are
closely
related
to
the
topics
discussed
in
§4.1
and
§4.2
—
between
various
components
of
the
proof
of
the
Mordell
Conjecture
given
in
[Falt1]
and
inter-universal
Teichmüller
theory:
flt
(1
)
Various
well-known
aspects
of
classical
algebraic
number
theory
related
to
the
“geometry
of
numbers”,
such
as
the
theory
of
heights
and
the
Hermite-Minkowski
theorem,
are
applied
in
[Falt1].
Similar
aspects
of
classical
algebraic
number
theory
may
be
seen
in
the
“non-interference”
property
[i.e.,
the
fact
that
the
only
nonzero
elements
of
an
NF
that
are
integral
at
all
nonarchimedean
and
archimedean
valua-
tions
of
the
NF
are
the
roots
of
unity]
for
copies
of
the
number
field
F
mod
discussed
in
§3.7,
(i),
as
well
as
in
the
use
of
global
realified
Frobenioids
associated
to
NF’s
[i.e.,
which
are
essentially
an
abstract
category-theoretic
version
of
the
classical
notions
of
arithmetic
degrees
and
heights].
flt
(2
)
Global
class
field
theory
for
NF’s,
as
well
as
the
closely
related
notion
of
Dirichlet
density
of
primes,
plays
an
important
role
in
[Falt1].
These
aspects
of
[Falt1]
are
compared
and
contrasted
in
substantial
detail
in
the
discussion
of
§4.2
with
the
Kummer
theory
that
plays
a
central
role
in
inter-universal
Teichmüller
theory.
flt
(3
)
The
theory
of
Hodge-Tate
decompositions
of
p-adic
Tate
modules
of
abelian
varieties
over
MLF’s
plays
an
important
role
both
in
[Falt1]
and,
as
discussed
in
cls
§4.1,
(ii),
(4
),
in
inter-universal
Teichmüller
theory.
158
Shinichi
Mochizuki
flt
(4
)
The
computations,
applied
in
[Falt1],
of
the
ramification
that
occurs
in
the
theory
surrounding
finite
flat
group
schemes
bear
a
rough
resemblance
to
the
ramifica-
tion
computations
involving
log-shells
in
[AbsTopIII];
[IUTchIV],
Propositions
1.1,
1.2,
1.3,
1.4.
flt
(5
)
The
hidden
endomorphisms
[cf.
the
discussion
of
[AbsTopII],
Introduction]
that
underlie
the
theory
of
Belyi
and
elliptic
cuspidalizations
[cf.
the
discussion
of
§3.3,
(vi);
§3.4,
(iii)],
which
play
an
important
role
in
inter-universal
Teichmüller
theory,
as
well
as
the
theory
of
noncritical
Belyi
maps
that
is
applied
[cf.
the
discussion
of
§3.7,
(iv)],
via
[GenEll],
§2,
in
[IUTchIV],
§2,
may
be
thought
of
as
a
sort
of
analogue
for
hyperbolic
curves
of
the
theory
of
isogenies
and
Tate
modules
of
abelian
varieties
that
plays
a
central
role
in
[Falt1].
flt
(6
)
The
important
role
played
by
polarizations
of
abelian
varieties
in
[Falt1]
may
be
compared
to
the
quite
central
role
played
by
commutators
of
theta
groups
in
the
theory
of
rigidity
properties
of
mono-theta
environments,
and
hence
in
inter-
gau
universal
Teichmüller
theory
as
a
whole
[cf.
the
discussion
of
§3.4,
(iv);
§3.8,
(9
);
cls
§4.1,
(ii),
(5
)].
flt
(7
)
The
logarithmic
geometry
of
toroidal
compactifications,
which
plays
an
important
role
in
[Falt1],
may
be
compared
to
the
logarithmic
geometry
of
special
fibers
of
stable
curves.
The
latter
instance
of
logarithmic
geometry
is
the
start-
ing
point
for
the
combinatorial
anabelian
geometry
of
tempered
fundamental
groups
developed
in
[Semi],
which
plays
an
important
role
throughout
inter-universal
Te-
ichmüller
theory.
(ii)
Arithmetic
holomorphicity
versus
mono-analyticity/multiradiality:
One
way
to
summarize
the
discussion
of
(i),
as
well
as
a
substantial
portion
of
the
dis-
cussion
of
§4.2
[cf.
[IUTchIV],
Remark
2.3.3,
(iii)],
is
as
follows:
inter-universal
Teichmüller
theory
may
be
understood,
to
a
substantial
extent,
as
a
sort
of
hyperbolic,
mono-analytic/multiradial
analogue
of
the
the
abelian,
arithmetic
holomorphic
theory
of
[Falt1].
Indeed,
this
is
precisely
the
point
of
view
of
the
discussion
of
§4.2,
(iii),
concerning
the
relationship
between
the
essentially
arithmetic
holomorphic
nature
of
global
class
field
theory
and
the
essentially
mono-analytic
nature
of
Kummer
theory.
If
one
takes
the
point
of
view
[cf.
the
discussion
of
§2.3,
§2.4,
§2.5,
§2.6;
Examples
2.14.2,
2.14.3]
that
Galois
or
arithmetic
fundamental
groups
should
be
thought
of
as
“arithmetic
tangent
bundles”,
then
the
point
of
view
of
the
present
discussion
may
be
formulated
in
the
following
way
[cf.
the
discussion
of
the
final
portion
of
[IUTchI],
§I2]:
Many
results
in
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
159
the
conventional
framework
of
arithmetic
geometry
that
concern
Galois
or
arithmetic
fundamental
groups
may
be
understood
as
results
to
the
effect
that
some
sort
of
“H
0
(arithmetic
tangent
bundle)”
does
indeed
coincide
with
some
sort
of
very
small
collection
of
scheme-theoretic
—
i.e.,
arithmetic
holomorphic
—
auto-/endo-morphisms.
Indeed,
examples
of
this
sort
of
phenomenon
include
(1
hol
(2
hol
(3
hol
(4
hol
)
the
version
of
the
Tate
Conjecture
proven
in
[Falt1];
)
various
bi-anabelian
results
[cf.
the
discussion
of
§2.7,
(v)]
—
i.e.,
fully
faith-
fulness
results
in
the
style
of
various
versions
of
the
“Grothendieck
Conjecture”
—
in
anabelian
geometry;
)
the
“tiny”
special
case
of
the
theory
of
[Falt1]
discussed
in
§2.3
to
the
effect
that
“Frobenius
endomorphisms
of
NF’s”
of
the
desired
type
[i.e.,
that
yield
bounds
on
heights
—
cf.
the
discussion
of
§2.4!]
cannot
exist,
i.e.,
so
long
as
one
restricts
oneself
to
working
within
the
framework
of
conventional
scheme
theory;
)
the
results
of
[Wiles]
concerning
Galois
representations
[cf.
the
discussion
of
[IUTchI],
§I5],
which
may
be
summarized
as
asserting,
in
essence,
that,
roughly
speaking,
nontrivial
deformations
of
Galois
representations
that
satisfy
suit-
able
natural
conditions
do
not
exist.
All
of
the
results
just
stated
assert
some
sort
of
“arithmetic
holomorphic
nonexistence”
[up
to
a
very
small
number
of
exceptions],
hence
lie
in
a
fundamentally
different
direction
from
the
content
of
inter-universal
Teichmüller
theory,
which,
in
effect,
concerns
the
construction
—
or
“non-arithmetic
holomorphic
existence”
—
of
a
“Frobenius
endomorphism
of
an
NF”,
by
working
outside
the
framework
of
conventional
scheme
theory,
i.e.,
by
considering
suitable
mono-analytic/multiradial
deformations
of
the
arithmetic
holomorphic
structure,
that
is
to
say,
at
the
level
of
suggestive
notation,
by
considering
“H
1
(arithmetic
tangent
bundle)”.
(iii)
Comparison
with
the
metric
proofs
of
Parshin
and
Bogomolov
in
the
complex
case:
Parshin
[cf.
[Par]]
and
Bogomolov
[cf.
[ABKP],
[Zh]]
have
given
proofs
of
geometric
versions
over
the
complex
numbers
of
the
Mordell
and
Szpiro
Conjectures,
respectively
[cf.
the
discussion
of
[IUTchIV],
Remarks
2.3.4,
2.3.5].
Parshin’s
proof
of
the
geometric
version
of
the
Mordell
Conjecture
is
discussed
Shinichi
Mochizuki
160
in
detail
in
[IUTchIV],
Remark
2.3.5,
while
Bogomolov’s
proof
of
the
geometric
version
of
the
Szpiro
Conjecture
is
discussed
in
detail
in
[BogIUT]
[cf.
also
§3.10,
(vi),
of
the
present
paper].
The
relationships
of
these
two
proofs
to
one
another,
as
well
as
to
the
arithmetic
theory,
may
be
summarized
as
follows:
(1
PB
(2
PB
(3
PB
(4
PB
(5
PB
)
Both
proofs
revolve
around
the
consideration
of
metric
estimates
of
the
dis-
placements
that
arise
from
various
natural
actions
of
elements
of
the
[usual
topo-
logical]
fundamental
groups
that
appear
[cf.
the
discussion
at
the
beginning
of
[IUTchIV],
Remark
2.3.5].
)
Both
Parshin’s
and
Bogomolov’s
proofs
concern
the
metric
geometry
of
the
complex
spaces
that
appear.
On
the
other
hand,
these
two
proofs
differ
fundamen-
tally
in
that
whereas
the
metric
geometry
that
appears
in
Parshin’s
proof
concerns
the
holomorphic
geometry
that
arises
from
the
Kobayashi
distance
—
i.e.,
in
effect,
the
Schwarz
lemma
of
elementary
complex
analysis
—
the
metric
geom-
etry
that
appears
in
Bogomolov’s
proof
concerns
the
real
analytic
hyperbolic
geometry
of
the
upper
half-plane
[cf.
[IUTchIV],
Remark
2.3.5,
(PB1)].
PB
)
The
difference
observed
in
(2
)
is
interesting
in
that
in
corresponds
precisely
to
the
difference
discussed
in
(ii)
above
between
the
proof
of
the
arithmetic
Mordell
Conjecture
[for
NF’s!]
in
[Falt1]
and
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
[IUTchIV],
Remark
2.3.5,
(i)].
)
Parshin’s
proof
concerns,
as
one
might
expect
from
the
statement
of
the
Mordell
Conjecture,
rough,
qualitative
estimates.
This
state
of
affairs
contrasts
sharply,
again
as
one
might
expect
from
the
statement
of
the
Szpiro
Conjecture,
with
Bogo-
molov’s
proof,
which
concerns
effective,
quantitative
estimates
[cf.
the
discus-
sion
of
[IUTchIV],
Remark
2.3.5,
(ii)].
)
The
appearance
of
the
Kobayashi
distance
—
i.e.,
in
essence,
the
Schwarz
lemma
PB
of
elementary
complex
analysis
—
in
(2
)
is
of
interest
in
light
of
the
point
of
view
discussed
in
§3.3,
(vi);
§3.7,
(iv),
concerning
the
correspondence
between
the
use
of
Belyi
maps
in
inter-universal
Teichmüller
theory,
i.e.,
in
the
context
of
Belyi
cuspidalizations
or
height
estimates,
as
a
sort
of
means
of
arithmetic
analytic
continuation,
and
the
classical
complex
theory
surrounding
the
Schwarz
lemma
[cf.
the
discussion
of
[IUTchIV],
Remark
2.3.5,
(iii)].
§
4.4.
Atavistic
resemblance
in
the
development
of
mathematics
(i)
Questioning
strictly
linear
models
of
evolution:
Progress
in
mathemat-
ics
is
often
portrayed
as
a
strictly
linear
affair
—
a
process
in
which
old
theories
or
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
161
ideas
are
rendered
essentially
obsolete,
and
hence
forgotten,
as
soon
as
the
essential
content
of
those
theories
or
ideas
is
“suitably
extracted/absorbed”
and
formulated
in
a
more
“modern
form”,
which
then
becomes
known
as
the
“state
of
the
art”.
The
his-
torical
development
of
mathematics
is
then
envisioned
as
a
sort
of
towering
edifice
that
is
subject
to
a
perpetual
appending
of
higher
and
higher
floors,
as
new
“states
of
the
art”
are
discovered.
On
the
other
hand,
it
is
often
overlooked
that
there
is
in
fact
no
intrinsic
justification
for
this
sort
of
strictly
linear
model
of
evolution.
Put
another
way,
there
is
no
rigorous
justification
for
excluding
the
possibility
that
a
particular
ap-
proach
to
mathematical
research
that
happens
to
be
embraced
without
doubt
by
a
particular
community
of
mathematicians
as
the
path
forward
in
this
sort
of
strictly
linear
evolutionary
model
may
in
fact
be
nothing
more
than
a
dramatic
“wrong
turn”,
i.e.,
a
sort
of
unproductive
march
into
a
meaningless
cul
de
sac.
Indeed,
Grothendieck’s
original
idea
that
anabelian
geometry
could
shed
light
on
diophantine
geometry
[cf.
the
discussion
at
the
beginning
of
[IUTchI],
§I5]
suggests
precisely
this
sort
of
skepticism
concerning
the
“linear
evolutionary
model”
that
arose
in
the
1960’s
to
the
effect
that
progress
in
arithmetic
geometry
was
best
understood
as
a
sort
of
strictly
linear
march
toward
the
goal
of
realizing
the
theory
of
motives,
i.e.,
a
sort
of
idealized
version
of
the
notion
of
a
Weil
cohomology.
In
more
recent
years,
another
major
“linear
evolutionary
model”
that
has
arisen,
partly
as
a
result
of
the
influence
of
the
work
of
Wiles
[cf.
[Wiles]]
concerning
Galois
repre-
sentations,
asserts
that
progress
in
arithmetic
geometry
is
best
understood
as
a
sort
of
strictly
linear
march
toward
the
goal
of
realizing
the
representation-theoretic
approach
to
arithmetic
geometry
constituted
by
the
Langlands
program.
As
discussed
in
§4.3,
(ii);
[IUTchI],
§I5,
(a
app
)
the
“mono-anabelian”
approach
to
arithmetic
geometry
constituted
by
inter-
universal
Teichmüller
theory
differs
fundamentally
not
only
from
(b
app
)
the
motive-/cohomology-theoretic
and
representation-theoretic
approaches
to
arith-
metic
geometry
just
discussed
—
both
of
which
may
be
characterized
as
“abelian”!
—
Shinichi
Mochizuki
162
but
also
from
(c
app
)
the
“bi-anabelian”
approach
involving
the
section
conjecture
that
was
ap-
parently
originally
envisioned
by
Grothendieck.
Here,
we
recall,
moreover,
the
point
of
view
of
the
dichotomy
discussed
in
§4.3,
(ii),
concerning
arithmetic
holomorphicity
and
mono-analyticity/multiradiality,
i.e.,
to
the
effect
that
the
difference
between
(a
app
),
on
the
one
hand,
and
both
(b
app
)
and
(c
app
),
on
the
other,
may
be
understood
[if,
for
the
sake
of
brevity,
one
applies
the
term
“holomorphic”
as
an
abbreviation
of
the
term
“arithmetically
holomorphic”]
as
the
difference
between
non-holomorphic
existence
and
holomorphic
nonexistence.
Another
way
to
understand,
at
a
very
rough
level,
the
difference
between
(a
app
)
and
(b
app
)
is
as
a
reflection
of
the
deep
structural
differences
between
[discrete
or
profinite]
free
groups
and
matrix
groups
[with
discrete
or
profi-
nite
coefficients]
—
cf.
the
discussion
of
[IUTchI],
§I5.
Finally,
at
a
much
more
elementary
level,
we
note
that
the
theory
of
Galois
groups
—
which
may
be
thought
of
as
a
mechanism
that
allows
one
to
pass
from
field
theory
to
group
theory
—
plays
a
fundamental
role
in
(a
app
),
(b
app
),
and
(c
app
).
From
this
point
of
view,
the
difference
between
(a
app
),
on
the
one
hand,
and
(b
app
)
[and,
to
a
slightly
lesser
extent,
(c
app
)],
on
the
other,
may
be
understood
as
the
difference
between
the
“inequalities”
group
theory
≫
field
theory
and
field
theory
≫
group
theory
—
i.e.,
the
issue
of
whether
one
regards
[abstract]
group
theory
as
the
central
object
of
interest,
while
field
theory
[which
we
understand
as
including
vector
spaces
over
fields,
hence
also
representation
theory]
is
relegated
to
playing
only
a
subordinate
role,
or
vice
versa.
In
this
context,
it
is
perhaps
of
interest
to
note
that
common
central
features
that
appear
in
both
inter-universal
Teichmüller
theory
and
the
work
of
Wiles
[cf.
[Wiles]]
concerning
Galois
representations
—
i.e.,
in
both
(a
app
)
and
(b
app
)
—
include
not
only
·
the
central
use
of
Galois
groups
[as
discussed
above],
but
also
·
the
central
use
of
function
theory
on
the
upper
half-plane,
i.e.,
theta
functions
in
the
case
of
inter-universal
Teichmüller
theory
and
modular
forms
in
the
case
of
[Wiles].
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
163
On
the
other
hand,
just
as
in
the
case
of
Galois
groups
discussed
above,
the
approaches
taken
in
inter-universal
Teichmüller
theory
and
[Wiles]
to
using
function
theory
on
the
upper
half-plane
—
i.e.,
theta
functions
versus
modular
forms
—
differ
quite
substan-
tially.
(ii)
Examples
of
atavistic
development:
An
alternative
point
of
view
to
the
sort
of
strictly
linear
evolutionary
model
discussed
in
(i)
is
the
point
of
view
that
progress
in
mathematics
is
best
understood
as
a
much
more
complicated
family
tree,
i.e.,
not
as
a
tree
that
consists
solely
of
a
single
trunk
without
branches
that
continues
to
grow
upward
in
a
strictly
linear
manner,
but
rather
as
a
much
more
complicated
organism,
whose
growth
is
sustained
by
an
intricate
mechanism
of
interaction
among
a
vast
multitude
of
branches,
some
of
which
sprout
not
from
branches
of
relatively
recent
vintage,
but
rather
from
much
older,
more
ancestral
branches
of
the
organism
that
were
entirely
irrelevant
to
the
recent
growth
of
the
organism.
In
the
context
of
the
present
paper,
it
is
of
interest
to
note
that
this
point
of
view,
i.e.,
of
substantially
different
multiple
evolutionary
branches
that
sprout
from
a
single
common
ancestral
branch,
is
reminiscent
of
the
notion
of
“mutually
alien
copies”,
which
forms
a
central
theme
of
the
present
paper
[cf.
the
dis-
cussion
of
§2.7,
(i),
(ii);
§3.8].
Phenomena
that
support
this
point
of
view
of
an
“atavistic
model
of
mathematical
development”
may
be
seen
in
many
of
the
examples
discussed
in
§4.1,
§4.2,
and
§4.3
such
as
the
following:
(1
atv
(2
atv
(3
atv
(4
atv
)
The
very
elementary
construction
of
Belyi
maps
in
the
early
1980’s,
or
indeed
noncritical
Belyi
maps
in
[NCBelyi],
could
easily
have
been
discovered
in
the
flt
PB
late
nineteenth
century
[cf.
§4.3,
(i),
(5
);
§4.3,
(iii),
(5
)].
)
The
application
of
Belyi
maps
to
Belyi
cuspidalization
[cf.
[AbsTopII],
§3]
atv
could
easily
have
been
discovered
in
the
mid-1990’s
[cf.
also
(1
)].
)
The
application
of
noncritical
Belyi
maps
to
height
estimates
in
[GenEll],
§2,
atv
could
easily
have
been
discovered
in
the
mid-1980’s
[cf.
also
(1
)].
)
The
Galois-theoretic
interpretation
of
the
Gaussian
integral
or
Jacobi’s
identity
furnished
by
inter-universal
Teichmüller
theory
[cf.
the
discussion
of
§3.8;
the
discussion
at
the
end
of
§3.9,
(iii);
the
discussion
of
the
final
portion
of
§4.1,
(i)]
could
easily
have
been
discovered
much
earlier
than
in
the
series
of
papers
[IUTchI],
[IUTchII],
[IUTchIII],
[IUTchIV].
Shinichi
Mochizuki
164
(5
atv
(6
atv
(7
atv
(8
atv
(9
atv
)
The
interpretation
of
changes
of
universe
in
the
context
of
non-ring-theoretic
“arithmetic
changes
of
coordinates”
as
in
the
discussion
of
§2.10
is
entirely
elemen-
cls
cls
tary
and
could
easily
have
been
discovered
in
the
1960’s
[cf.
§4.1,
(ii),
(1
),
(2
)].
)
The
use
of
Hodge-Tate
representations
as
in
[Q
p
GC]
or
[AbsTopI],
§3,
could
cls
easily
have
been
discovered
in
the
1960’s
[cf.
§4.1,
(ii),
(3
)].
)
The
use
of
Hodge-Tate
decompositions
as
in
[pGC]
could
easily
have
been
cls
discovered
in
the
1980’s
[cf.
§4.1,
(ii),
(4
)].
)
The
anabelian
approach
to
theta
functions
on
Tate
curves
taken
in
[EtTh]
[cf.
§3.4,
(iii),
(iv)]
could
easily
have
been
discovered
in
the
mid-1990’s
[cf.
§4.1,
cls
(ii),
(5
)].
)
The
non-representation-theoretic
use
of
the
structure
of
theta
groups
in
the
theory
of
[EtTh]
[cf.
the
discussion
at
the
end
of
§3.4,
(iv)]
could
easily
have
cls
cls
been
discovered
in
the
1980’s
[cf.
§4.1,
(ii),
(5
),
(6
)].
(10
atv
(11
atv
(12
atv
(13
atv
(14
atv
)
Scheme-theoretic
Hodge-Arakelov
theory,
which
may
be
regarded
as
a
natural
extension
of
the
[the
portion
concerning
elliptic
curves
of]
Mumford’s
theory
of
algebraic
theta
functions,
could
easily
have
been
discovered
in
the
late
1960’s
cls
[cf.
§4.1,
(ii),
(6
)].
)
The
technique
of
mono-anabelian
transport
in
the
context
of
positive
char-
acteristic
anabelian
geometry,
i.e.,
in
the
style
of
Example
2.6.1,
could
easily
have
cls
been
discovered
in
the
1980’s
[cf.
§4.1,
(ii),
(7
)].
)
The
use
of
monoids
as
in
the
theory
of
Frobenioids
could
easily
have
been
cls
discovered
in
the
mid-1990’s
[cf.
§4.1,
(ii),
(8
)].
)
The
notion
of
multiradiality
is
entirely
elementary
and
could
easily
have
been
cls
discovered
in
the
late
1960’s
[cf.
§4.1,
(ii),
(9
)].
)
The
point
of
view
of
taking
a
Kummer-theoretic
approach
to
Kronecker’s
Jugendtraum,
i.e.,
as
discussed
in
§4.2,
(ii),
could
easily
have
been
discovered
much
earlier
than
in
the
series
of
papers
[IUTchI],
[IUTchII],
[IUTchIII],
[IUTchIV].
In
this
context,
we
note
that
the
atavistic
model
of
mathematical
development
just
discussed
also
suggests
the
possibility
that
the
theory
of
Frobenioids
—
which,
as
was
discussed
in
§3.5,
was
originally
developed
for
reasons
that
were
[related
to,
but,
strictly
speaking]
independent
of
inter-universal
Teichmüller
theory,
and
is,
in
fact,
only
used
in
inter-universal
Teichmüller
theory
in
a
relatively
weak
sense
—
may
give
rise,
at
some
distant
future
date,
to
further
developments
of
interest
that
are
not
directly
related
to
inter-universal
Teichmüller
theory.
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
165
(iii)
Escaping
from
the
cage
of
deterministic
models
of
mathematical
development:
The
adoption
of
strictly
linear
evolutionary
models
of
progress
in
mathematics
of
the
sort
discussed
in
(i)
tends
to
be
highly
attractive
to
many
math-
ematicians
in
light
of
the
intoxicating
simplicity
of
such
strictly
linear
evolutionary
models,
by
comparison
to
the
more
complicated
point
of
view
discussed
in
(ii).
This
intoxicating
simplicity
also
makes
such
strictly
linear
evolutionary
models
—
together
with
strictly
linear
numerical
evaluation
devices
such
as
the
“number
of
papers
pub-
lished”,
the
“number
of
citations
of
published
papers”,
or
other
like-minded
narrowly
defined
data
formats
that
have
been
concocted
for
measuring
progress
in
mathematics
—
highly
enticing
to
administrators
who
are
charged
with
the
tasks
of
evaluating,
hir-
ing,
or
promoting
mathematicians.
Moreover,
this
state
of
affairs
that
regulates
the
collection
of
individuals
who
are
granted
the
license
and
resources
necessary
to
actively
engage
in
mathematical
research
tends
to
have
the
effect,
over
the
long
term,
of
stifling
efforts
by
young
researchers
to
conduct
long-term
mathematical
research
in
di-
rections
that
substantially
diverge
from
the
strictly
linear
evolutionary
models
that
have
been
adopted,
thus
making
it
exceedingly
difficult
for
new
“unanticipated”
evolutionary
branches
in
the
development
of
mathematics
to
sprout.
Put
another
way,
inappropriately
narrowly
defined
strictly
linear
evolutionary
models
of
progress
in
mathematics
exhibit
a
strong
and
unfortunate
tendency
in
the
pro-
fession
of
mathematics
as
it
is
currently
practiced
to
become
something
of
a
self-fulfilling
prophecy
—
a
“prophecy”
that
is
often
zealously
rationalized
by
dubious
bouts
of
circular
reasoning.
In
particular,
the
issue
of
escaping
from
the
cage
of
such
narrowly
defined
deterministic
models
of
mathematical
development
stands
out
as
an
issue
of
crucial
strategic
impor-
tance
from
the
point
of
view
of
charting
a
sound,
sustainable
course
in
the
future
development
of
the
field
of
mathematics,
i.e.,
a
course
that
cherishes
the
priviledge
to
foster
genuinely
novel
and
unforeseen
evolutionary
branches
in
its
development.
Bibliography
[ABKP]
J.
Amorós,
F.
Bogomolov,
L.
Katzarkov,
T.
Pantev,
Symplectic
Lefshetz
fibration
with
arbitrary
fundamental
groups,
J.
Differential
Geom.
54
(2000),
pp.
489-545.
[André]
Y.
André,
On
a
Geometric
Description
of
Gal(Q
p
/Q
p
)
and
a
p-adic
Avatar
of
GT
,
Duke
Math.
J.
119
(2003),
pp.
1-39.
[Bell]
D.
Bell,
Poisson’s
remarkable
calculation
—
a
method
or
a
trick?,
Elem.
Math.
65
(2010),
pp.
29-36.
166
Shinichi
Mochizuki
[Dawson]
R.
J.
MacG.
Dawson,
On
a
“singular”
integration
technique
of
Poisson,
Amer.
Math.
Monthly
112
(2005),
pp.
270-272.
[Falt1]
G.
Faltings,
Endlichkeitssätze
für
Abelschen
Varietäten
über
Zahlkörpern,
Invent.
Math.
73
(1983),
pp.
349-366.
[Falt2]
G.
Faltings,
p-adic
Hodge
Theory,
Journal
of
the
Amer.
Math.
Soc.
1
(1988),
pp.
255-299.
[Fsk]
I.
Fesenko,
Arithmetic
deformation
theory
via
arithmetic
fundamental
groups
and
nonarchimedean
theta-functions,
notes
on
the
work
of
Shinichi
Mochizuki,
Eur.
J.
Math.
1
(2015),
pp.
405-440.
[GrCrs]
A.
Grothendieck,
Crystals
and
the
de
Rham
cohomology
of
schemes.
Notes
by
I.
Coates
and
O.
Jussila,
in
Dix
exposés
sur
la
cohomologie
des
schémas,
Adv.
Stud.
Pure
Math.
3,
North-Holland
(1968),
pp.
306-358.
[Harts]
R.
Hartshorne,
Algebraic
Geometry,
Graduate
Texts
in
Math.
52,
Springer-Verlag
(1997).
[NodNon]
Y.
Hoshi,
S.
Mochizuki,
On
the
Combinatorial
Anabelian
Geometry
of
Nodally
Nondegenerate
Outer
Representations,
Hiroshima
Math.
J.
41
(2011),
pp.
275-342.
[CbTpII]
Y.
Hoshi,
S.
Mochizuki,
Topics
Surrounding
the
Combinatorial
Anabelian
Geometry
of
Hyperbolic
Curves
II:
Tripods
and
Combinatorial
Cuspidalization,
RIMS
Preprint
1762
(November
2012).
[Kato1]
K.
Kato,
Logarithmic
Structures
of
Fontaine-Illusie,
Algebraic
analysis,
geometry,
and
number
theory
(Baltimore,
MD,
1988),
Johns
Hopkins
Univ.
Press
(1989),
pp.
191-224.
[Kato2]
K.
Kato,
Toric
Singularities,
Amer.
J.
Math.
116
(1994),
pp.
1073-1099.
[KKMS]
G.
Kempf,
F.
F.
Knudsen,
D.
Mumford,
B.
Saint-Donat,
Toroidal
embeddings.
I.,
Lecture
Notes
in
Mathematics
339,
Springer-Verlag
(1973).
[Kobl]
N.
Koblitz,
p-adic
Numbers,
p-adic
Analysis,
and
Zeta
Functions,
Graduate
Texts
in
Mathematics
58,
Springer-Verlag
(1977).
[Mss]
D.
W.
Masser,
Note
on
a
conjecture
of
Szpiro
in
Astérisque
183
(1990),
pp.
19-23.
[MlWd]
D.
V.
Mathews,
The
hyperbolic
meaning
of
the
Milnor-Wood
inequality,
Expo.
Math.
30
(2012),
pp.
49-68.
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
167
[pOrd]
S.
Mochizuki,
A
Theory
of
Ordinary
p-adic
Curves,
Publ.
Res.
Inst.
Math.
Sci.
32
(1996),
pp.
957-1151.
[PrfGC]
S.
Mochizuki,
The
Profinite
Grothendieck
Conjecture
for
Closed
Hyperbolic
Curves
over
Number
Fields,
J.
Math.
Sci.
Univ.
Tokyo
3
(1996),
pp.
571-627.
[Q
p
GC]
S.
Mochizuki,
A
Version
of
the
Grothendieck
Conjecture
for
p-adic
Local
Fields,
The
International
Journal
of
Math.
8
(1997),
pp.
499-506.
[pTch]
S.
Mochizuki,
Foundations
of
p-adic
Teichmüller
Theory,
AMS/IP
Studies
in
Ad-
vanced
Mathematics
11,
American
Mathematical
Society/International
Press
(1999).
[pGC]
S.
Mochizuki,
The
Local
Pro-p
Anabelian
Geometry
of
Curves,
Invent.
Math.
138
(1999),
pp.
319-423.
[HASurI]
S.
Mochizuki,
A
Survey
of
the
Hodge-Arakelov
Theory
of
Elliptic
Curves
I,
Arith-
metic
Fundamental
Groups
and
Noncommutative
Algebra,
Proceedings
of
Symposia
in
Pure
Mathematics
70,
American
Mathematical
Society
(2002),
pp.
533-569.
[HASurII]
S.
Mochizuki,
A
Survey
of
the
Hodge-Arakelov
Theory
of
Elliptic
Curves
II,
Al-
gebraic
Geometry
2000,
Azumino,
Adv.
Stud.
Pure
Math.
36,
Math.
Soc.
Japan
(2002),
pp.
81-114.
[pTchIn]
S.
Mochizuki,
An
Introduction
to
p-adic
Teichmüller
Theory,
Cohomologies
p-adiques
et
applications
arithmétiques
I,
Astérisque
278
(2002),
pp.
1-49.
[CanLift]
S.
Mochizuki,
The
Absolute
Anabelian
Geometry
of
Canonical
Curves,
Kazuya
Kato’s
fiftieth
birthday,
Doc.
Math.
2003,
Extra
Vol.,
pp.
609-640.
[GeoAnbd]
S.
Mochizuki,
The
Geometry
of
Anabelioids,
Publ.
Res.
Inst.
Math.
Sci.
40
(2004),
pp.
819-881.
[AbsAnab]
S.
Mochizuki,
The
Absolute
Anabelian
Geometry
of
Hyperbolic
Curves,
Galois
The-
ory
and
Modular
Forms,
Kluwer
Academic
Publishers
(2004),
pp.
77-122.
[NCBelyi]
S.
Mochizuki,
Noncritical
Belyi
Maps,
Math.
J.
Okayama
Univ.
46
(2004),
pp.
105-
113.
[QuCnf]
S.
Mochizuki,
Conformal
and
quasiconformal
categorical
representation
of
hyper-
bolic
Riemann
surfaces,
Hiroshima
Math.
J.
36
(2006),
pp.
405-441.
[Semi]
S.
Mochizuki,
Semi-graphs
of
Anabelioids,
Publ.
Res.
Inst.
Math.
Sci.
42
(2006),
pp.
221-322.
168
Shinichi
Mochizuki
[Cusp]
S.
Mochizuki,
Absolute
anabelian
cuspidalizations
of
proper
hyperbolic
curves,
J.
Math.
Kyoto
Univ.
47
(2007),
pp.
451-539.
[MT]
S.
Mochizuki,
A.
Tamagawa,
The
algebraic
and
anabelian
geometry
of
configuration
spaces,
Hokkaido
Math.
J.
37
(2008),
pp.
75-131.
[FrdI]
S.
Mochizuki,
The
Geometry
of
Frobenioids
I:
The
General
Theory,
Kyushu
J.
Math.
62
(2008),
pp.
293-400.
[FrdII]
S.
Mochizuki,
The
Geometry
of
Frobenioids
II:
Poly-Frobenioids,
Kyushu
J.
Math.
62
(2008),
pp.
401-460.
[EtTh]
S.
Mochizuki,
The
Étale
Theta
Function
and
its
Frobenioid-theoretic
Manifesta-
tions,
Publ.
Res.
Inst.
Math.
Sci.
45
(2009),
pp.
227-349.
[GenEll]
S.
Mochizuki,
Arithmetic
Elliptic
Curves
in
General
Position,
Math.
J.
Okayama
Univ.
52
(2010),
pp.
1-28.
[CombCusp]
S.
Mochizuki,
On
the
Combinatorial
Cuspidalization
of
Hyperbolic
Curves,
Osaka
J.
Math.
47
(2010),
pp.
651-715.
[AbsTopI]
S.
Mochizuki,
Topics
in
Absolute
Anabelian
Geometry
I:
Generalities,
J.
Math.
Sci.
Univ.
Tokyo
19
(2012),
pp.
139-242.
[AbsTopII]
S.
Mochizuki,
Topics
in
Absolute
Anabelian
Geometry
II:
Decomposition
Groups
and
Endomorphisms,
J.
Math.
Sci.
Univ.
Tokyo
20
(2013),
pp.
171-269.
[Pano]
S.
Mochizuki,
A
Panoramic
Overview
of
Inter-universal
Teichmüller
Theory,
Alge-
braic
number
theory
and
related
topics
2012,
RIMS
Kôkyûroku
Bessatsu
B51,
Res.
Inst.
Math.
Sci.
(RIMS),
Kyoto
(2014),
pp.
301-345.
[AbsTopIII]
S.
Mochizuki,
Topics
in
Absolute
Anabelian
Geometry
III:
Global
Reconstruction
Algorithms,
J.
Math.
Sci.
Univ.
Tokyo
22
(2015),
pp.
939-1156.
[BogIUT]
S.
Mochizuki,
Bogomolov’s
proof
of
the
geometric
version
of
the
Szpiro
Conjecture
from
the
point
of
view
of
inter-universal
Teichmüller
theory,
Res.
Math.
Sci.
3
(2016),
3:6.
[IUTchI]
S.
Mochizuki,
Inter-universal
Teichmüller
Theory
I:
Construction
of
Hodge
The-
aters,
RIMS
Preprint
1756
(August
2012),
to
appear
in
Publ.
Res.
Inst.
Math.
Sci.
57
(2021).
[IUTchII]
S.
Mochizuki,
Inter-universal
Teichmüller
Theory
II:
Hodge-Arakelov-theoretic
Eval-
uation,
RIMS
Preprint
1757
(August
2012),
to
appear
in
Publ.
Res.
Inst.
Math.
Sci.
57
(2021).
Alien
copies,
Gaussians,
&
Inter-universal
Teichmüller
theory
169
[IUTchIII]
S.
Mochizuki,
Inter-universal
Teichmüller
Theory
III:
Canonical
Splittings
of
the
Log-theta-lattice,
RIMS
Preprint
1758
(August
2012),
to
appear
in
Publ.
Res.
Inst.
Math.
Sci.
57
(2021).
[IUTchIV]
S.
Mochizuki,
Inter-universal
Teichmüller
Theory
IV:
Log-volume
Computations
and
Set-theoretic
Foundations,
RIMS
Preprint
1759
(August
2012),
to
appear
in
Publ.
Res.
Inst.
Math.
Sci.
57
(2021).
[Mumf1]
D.
Mumford,
On
the
equations
defining
abelian
varieties
I,
II,
III,
Inv.
Math.
1
(1966),
pp.
287-354;
2
(1967),
pp.
71-135;
3
(1967),
pp.
215-244.
[Mumf2]
D.
Mumford,
An
Analytic
Construction
of
Degenerating
Abelian
Varieties
over
Complete
Rings,
Appendix
to
G.
Faltings
and
C.-L.
Chai,
Degenerations
of
Abelian
Varieties,
Springer-Verlag
(1990).
[NSW]
J.
Neukirch,
A.
Schmidt,
K.
Wingberg,
Cohomology
of
number
fields,
Grundlehren
der
Mathematischen
Wissenschaften
323,
Springer-Verlag
(2000).
[Par]
A.
N.
Parshin,
Finiteness
theorems
and
hyperbolic
manifolds,
The
Grothendieck
Festschrift,
Vol.
III,
Progress
in
Mathematics
88,
Birkhäuser
(1990).
[PopBog]
F.
Pop,
On
the
birational
anabelian
program
initiated
by
Bogomolov
I,
Invent.
Math.
187
(2012),
pp.
511-533.
[Serre]
J.-P.
Serre,
Abelian
l-adic
Representations
and
Elliptic
Curves,
W.
A.
Benjamin,
Inc.
(1968).
[SGA1]
A.
Grothendieck
et
al.,
Revêtement
étales
et
groupe
fondamental,
Séminaire
de
Géometrie
Algébrique
du
Bois
Marie
1960-1961
(SGA1),
dirigé
par
A.
Grothendieck,
augmenté
de
deux
exposés
de
M.
Raynaud,
Lecture
Notes
in
Mathematics
224,
Springer-Verlag
(1971).
[SGA4]
A.
Grothendieck
et
al.,
Théorie
des
topos
et
cohomologie
étale,
Lecture
Notes
in
Mathematics
264,
270,
305,
Springer-Verlag
(1972-3).
[StYu]
C.
L.
Stewart,
K.
Yu,
On
the
abc
conjecture
II,
Duke
Math.
J.
108
(2001),
pp.
169-181.
[Uchi]
K.
Uchida,
Isomorphisms
of
Galois
groups
of
algebraic
function
fields,
Ann.
of
Math.
106
(1977),
pp.
589-598.
[vFr]
M.
van
Frankenhuijsen,
About
the
ABC
conjecture
and
an
alternative,
Number
theory,
analysis
and
geometry,
Springer-Verlag
(2012),
pp.
169-180.
170
Shinichi
Mochizuki
[Wada]
Y.
Wada,
Near
miss
abc-triples
in
compactly
bounded
subsets,
RIMS
Preprint
1855
(July
2016).
[Wiles]
A.
Wiles,
Modular
elliptic
curves
and
Fermat’s
last
theorem,
Ann.
of
Math.
141
(1995),
pp.
443-551.
[Zh]
S.
Zhang,
Geometry
of
algebraic
points,
First
International
Congress
of
Chinese
Mathematicians
(Beijing,
1998),
AMS/IP
Stud.
Adv.
Math.
20
(2001),
pp.
185-
198.
Updated
versions
of
preprints
by
the
author
are
available
at
the
following
webpage:
http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html